#7624334

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	#62324	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1238 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)(19,27)(20,31)(21,33)(23,36)(25,40)(26,39)(28,43)(29,45)(30,44)(32,35)(34,46)(37,47)(38,48)(41,42);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,24)(23,30)(27,43)(29,42)(31,40)(32,47)(33,39)(34,38)(35,37)(36,44)(41,45)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,23)(20,32)(21,35)(22,30)(24,28)(25,41)(26,42)(27,31)(29,33)(34,45)(36,40)(37,39)(38,47)(43,48)(44,46);(3,11)(6,9)(10,16)(13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,36)(28,30)(29,37)(31,40)(32,41)(33,39)(34,38)(35,42)(43,44)(45,47)(46,48);(2,8)(6,9)(7,14)(13,15)(18,24)(23,28)(25,38)(26,39)(36,43)(37,42)(40,48)(41,47)

Polycyclic presentation:

2^9|c^2=f,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, 7531530);