#8195559

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312126, #72312132, #72312174, …
maximal normal subgroup of	#72312126, #72312132, #72312174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8102755);