#7904208

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

4886 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811404);