#8422191

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	#36581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5094 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329387);