#7688112

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

4768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595308);