#8127451

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

4278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,4)(6,7)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(26,30)(31,33)(32,37)(38,39);(1,3,5,4)(2,6,8,7)(11,18)(12,17)(19,24)(20,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,5,3)(2,7,8,6)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034647);