#8460728

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

4894 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,31,30);(3,8)(4,6)(10,14)(11,16)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(9,12)(10,11)(13,15)(14,16)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,27,31,23);(1,2)(3,6)(4,8)(5,7)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,27,29,23)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,e^2=hi,f^2=hi,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, 8367924);