#8445503

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

5010 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=ghi,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, 8352699);