#8456832

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

5095 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)(3,8)(4,6)(5,7)(10,15)(11,12)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28);(3,4)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32);(1,3,5,4)(2,6,7,8)(9,11,14,12)(10,16,15,13)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29);(1,4,5,3)(2,6,7,8)(9,12,14,11)(10,16,15,13)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29);(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28);(1,2)(3,6)(4,8)(5,7)(9,13,14,16)(10,12,15,11)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8364028);