#8709104

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4982 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)(3,7)(4,8)(5,6)(10,13)(11,15)(17,18)(19,25)(20,23)(21,26)(22,28)(24,29)(27,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,29)(18,23,26,24)(19,27,28,32)(22,30,25,31);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,29)(22,31)(23,26)(25,30)(28,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,25)(20,27)(21,28)(22,26)(23,31)(24,30)(29,32);(1,3)(2,7)(4,6)(5,8)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,30)(31,32)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8616300);