#7624341

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,25)(19,29)(21,24)(23,37)(26,41)(28,34)(30,43)(31,45)(32,47)(33,46)(35,36)(38,39)(40,44)(42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,22,25)(19,30,33,26)(20,24,27,21)(23,28,36,40)(29,41,46,43)(31,32,42,39)(34,37,44,35)(38,48,47,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,33)(18,26,25,30)(20,31,27,42)(21,32,24,39)(23,38,36,47)(28,45,40,48)(29,35,46,37)(34,41,44,43);(3,11)(6,9)(10,16)(13,15)(17,20,22,27)(18,21,25,24)(19,31,33,42)(23,34,36,44)(26,32,30,39)(28,35,40,37)(29,45,46,48)(38,41,47,43);(2,8)(6,9)(7,14)(13,15)(17,21,22,24)(18,27,25,20)(19,32,33,39)(23,35,36,37)(26,42,30,31)(28,44,40,34)(29,38,46,47)(41,45,43,48)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531537);