#7624426

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
prime radical	#33679	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1179 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)(17,18)(19,24)(20,27)(21,35)(22,26)(23,29)(25,31)(28,37)(30,40)(32,42)(33,39)(34,46)(36,41)(38,45)(43,48)(44,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,33)(21,31)(22,30)(23,32)(25,35)(26,40)(27,39)(28,41)(29,42)(34,48)(36,37)(38,47)(43,46)(44,45);(17,19)(18,25)(20,30)(21,36)(22,33)(23,32)(24,35)(26,43)(27,41)(28,39)(29,44)(31,37)(34,47)(38,48)(40,46)(42,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,22)(18,26,29,27)(19,30,32,33)(21,34,38,37)(24,39,42,40)(25,43,44,41)(28,45,46,35)(31,36,47,48);(2,8)(6,9)(7,16)(14,15)(17,20,23,22)(18,27,29,26)(19,30,32,33)(21,34,38,37)(24,40,42,39)(25,41,44,43)(28,35,46,45)(31,36,47,48)

Polycyclic presentation:

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

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, 7531622);