#7683811

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591007);