#7855822

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

4997 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7763018);