#7624944

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1209 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)(19,28)(20,25)(21,36)(22,35)(23,27)(24,32)(26,37)(29,42)(30,41)(31,43)(33,48)(34,45)(38,39)(40,46)(44,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,29)(20,35)(21,26)(22,25)(23,27)(24,34)(28,42)(30,39)(31,40)(32,45)(33,44)(36,37)(38,41)(43,46)(47,48);(17,19,30,46,22,33,21,32,23,34,37,47,20,31,38,42)(18,24,39,48,25,40,26,28,27,29,36,43,35,44,41,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,20,23,22)(18,25,27,35)(19,31,34,33)(21,30,38,37)(24,40,29,44)(26,36,41,39)(28,43,45,48)(32,46,42,47)

Polycyclic presentation:

2^9|b^2=f,c^2=ghi,d^2=f,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

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, 7532140);