#7020702

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308943, #72308949
maximal normal subgroup of	#72308943, #72308949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6927898);