#8457066

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

5051 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8364262);