#508109

order	512
nilpotency class	4
isoclinism	isoclinic class of #508104
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29636
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#585	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

785 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,17,38)(4,18,22,52)(5,10,24,37)(6,25,28,11)(8,26,31,16)(9,32,35,21)(14,42,46,64)(15,41,48,63)(19,34,53,39)(20,33,27,51)(23,55,57,36)(29,58,61,43)(30,56,62,47)(40,54,59,45)(44,50,49,60);(1,3,7,17)(2,8,12,31)(4,14,22,46)(5,15,24,48)(6,26,28,16)(9,29,35,61)(10,30,37,62)(11,38,25,13)(18,50,52,60)(19,43,53,58)(20,49,27,44)(21,54,32,45)(23,56,57,47)(33,42,51,64)(34,40,39,59)(36,63,55,41);(1,4)(2,9)(3,14)(5,21)(6,27)(7,22)(8,29)(10,18)(11,39)(12,35)(13,40)(15,45)(16,49)(17,46)(19,23)(20,28)(24,32)(25,34)(26,44)(30,60)(31,61)(33,36)(37,52)(38,59)(41,42)(43,47)(48,54)(50,62)(51,55)(53,57)(56,58)(63,64);(1,5,7,24)(2,10,12,37)(3,15,17,48)(4,19)(6,23,28,57)(8,30,31,62)(9,33)(11,36,25,55)(13,41,38,63)(14,43)(16,47,26,56)(18,34)(20,21)(22,53)(27,32)(29,42)(35,51)(39,52)(40,50)(44,45)(46,58)(49,54)(59,60)(61,64)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gh,c^b=cg,e^b=ei,f^b=fh,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415305);