#7487390

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394586);