#8496570

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

4800 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)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12)(18,19)(20,27)(24,26)(28,32);(9,11)(10,12)(13,16)(14,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(2,4)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32);(1,4,5,2)(3,7,8,6)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,f^2=gi,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, 8403766);