#7715108

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

4665 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622304);