#8422161

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4894 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,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, 8329357);