#8926148

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

4981 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833344);