#506237

order	512
nilpotency class	4
isoclinism	isoclinic class of #499903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28117
prime radical	#33530	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,8)(4,18)(5,23)(6,12)(7,27)(9,21)(10,26)(11,13)(14,40)(15,43)(16,32)(17,47)(19,50)(20,49)(22,53)(24,37)(25,54)(28,35)(29,41)(30,46)(31,38)(33,52)(34,56)(36,39)(42,63)(44,61)(45,64)(48,59)(51,57)(55,58)(60,62);(1,3,13,38,6,16,7,17)(2,8,11,31,12,32,27,47)(4,19,22,48,20,14,28,42)(5,15,39,62,25,45,26,46)(9,29,37,61,33,57,34,58)(10,30,23,43,36,60,54,64)(18,40,53,63,49,50,35,59)(21,51,24,55,52,41,56,44);(1,4)(2,9)(3,14)(5,24)(6,20)(7,28)(8,29)(10,35)(11,37)(12,33)(13,22)(15,44)(16,19)(17,48)(18,23)(21,26)(25,56)(27,34)(30,59)(31,61)(32,57)(36,53)(38,42)(39,52)(40,43)(41,46)(45,55)(47,58)(49,54)(50,64)(51,62)(60,63);(5,25)(10,36)(15,45)(18,49)(21,52)(23,54)(24,56)(26,39)(30,60)(35,53)(40,50)(41,51)(43,64)(44,55)(46,62)(59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413433);