#7623430

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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	#60707	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1237 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)(18,24)(19,29)(20,23)(22,35)(25,39)(27,42)(30,41)(31,47)(32,46)(33,34)(36,37)(38,44)(40,43)(45,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,21,24)(19,30,32,25)(20,26,23,28)(22,27,34,38)(29,39,46,41)(31,45,37,40)(33,42,35,44)(36,48,47,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,32)(18,25,24,30)(20,31,23,37)(22,36,34,47)(26,40,28,45)(27,43,38,48)(29,33,46,35)(39,44,41,42);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,26,24,28)(19,31,32,37)(22,33,34,35)(25,40,30,45)(27,42,38,44)(29,36,46,47)(39,48,41,43);(18,24)(25,30)(26,28)(27,38)(39,41)(40,45)(42,44)(43,48)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^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=cf,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530626);