#8179704

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

5100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086900);