#7795835

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

4990 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)(9,10,13,14)(11,12,15,16)(17,18,23,20)(19,21,25,26)(22,24,30,28)(27,29,31,32);(1,2,5,6)(3,7,4,8)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31);(10,14)(11,15)(18,20)(19,25)(22,30)(29,32);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,15)(14,16)(17,20,23,18)(19,26,25,21)(22,28,30,24)(27,32,31,29);(9,12)(10,11)(13,16)(14,15)(17,21,23,26)(18,19,20,25)(22,29,30,32)(24,27,28,31);(1,4,5,3)(2,8,6,7)(17,22)(18,24)(19,27)(20,28)(21,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7703031);