#8458811

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

5046 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)(9,10,13,16)(11,12,15,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(18,22)(23,25)(24,29)(30,31);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(2,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4,5,3)(2,8,7,6)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^2=ghi,f^2=i,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, 8366007);