#8472886

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

5418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380082);