#7623441

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1226 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)(21,34)(22,32)(23,27)(24,31)(26,36)(29,44)(30,45)(33,43)(35,40)(37,47)(38,42)(39,46)(41,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,29)(24,39)(28,44)(30,38)(31,46)(35,48)(40,41)(42,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,24)(20,30)(21,35)(22,38)(23,29)(25,40)(26,42)(27,39)(28,34)(31,36)(32,41)(33,46)(37,48)(43,45)(44,47);(2,8)(6,9)(7,16)(14,15)(17,18)(19,24)(20,32)(21,26)(22,25)(23,27)(28,31)(29,39)(30,41)(33,47)(34,36)(35,42)(37,43)(38,40)(44,46)(45,48);(17,20,23,22)(18,25,27,32)(19,30,29,38)(21,33,37,36)(24,40,39,41)(26,34,43,47)(28,45,44,42)(31,35,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530637);