#8757077

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

4526 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664273);