#7842431

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

4687 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7749627);