#512041

order	512
nilpotency class	4
isoclinism	isoclinic class of #511841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32976
prime radical	#3529	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

735 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=di,d^c=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419237);