#7726804

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
prime radical	#35358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4879 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7634000);