#7594219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44922	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(18,23)(19,27)(20,21)(24,31)(25,26)(28,29);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,21)(25,26)(27,29)(30,32);(1,3,5,4)(2,7,6,8)(17,18,21,26,22,25,20,23)(19,24,28,32,29,31,27,30);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,19,21,28,22,29,20,27)(18,24,26,32,25,31,23,30);(1,4,5,3)(2,7,6,8)(18,25)(23,26)(24,31)(30,32)

Polycyclic presentation:

2^9|a^2=fgh,c^2=fgi,d^2=fgi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7501415);