#7657101

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	#36020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4647 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564297);