#8094942

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

4677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 8002138);