#7624338

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

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^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,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, 7531534);