#8505958

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

5015 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)(11,15)(12,16)(13,14)(20,28)(24,29)(27,32)(30,31);(1,3,5,8)(2,6,4,7)(10,14)(11,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32);(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31);(2,4)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31);(1,4,5,2)(3,7,8,6)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=hi,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, 8413154);