#8152158

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

4893 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059354);