#8664518

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

4827 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571714);