#7804407

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

4747 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)(19,26)(20,25)(21,23)(22,27)(24,30)(28,31)(29,32);(1,2,5,6)(3,7,4,8)(17,19)(18,23)(20,29)(21,22)(24,32)(25,31)(26,27)(28,30);(10,11)(14,15)(17,20,22,30)(18,24,27,25)(19,28,21,29)(23,31,26,32);(1,3,5,4)(2,7,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,25)(19,29)(21,28)(22,30)(23,31)(24,27)(26,32);(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,26,27,23)(20,29,30,28)(24,32,25,31);(1,4,5,3)(2,8,6,7)(17,18)(19,23)(20,25)(21,26)(22,27)(24,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=gi,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, 7711603);