#8457051

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

5008 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)(3,8)(4,6)(5,7)(9,10,13,11)(12,15,16,14)(18,23)(19,26)(25,30)(27,32);(1,3,5,4)(2,6,7,8)(10,11)(14,15)(17,18,22,23)(19,21,26,24)(20,25,29,30)(27,28,32,31);(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,27)(22,26)(23,24)(25,28)(29,32)(30,31);(2,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32);(1,4,5,3)(2,8,7,6)(17,20,22,29)(18,25,23,30)(19,27,26,32)(21,28,24,31);(1,2)(3,6)(4,8)(5,7)(17,21,22,24)(18,26,23,19)(20,28,29,31)(25,32,30,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=gi,f^2=h,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, 8364247);