#8799089

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

5047 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706285);