#7624371

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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	#62251	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1277 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)(17,18,21,27)(19,26,23,24)(20,31,32,47)(22,35,34,40)(25,41,43,33)(28,30,44,37)(29,46,38,45)(36,42,48,39);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,23)(18,24,27,26)(20,29,32,38)(22,30,34,37)(25,39,43,42)(28,40,44,35)(31,45,47,46)(33,36,41,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,32)(18,25,27,43)(19,29,23,38)(22,36,34,48)(24,39,26,42)(28,45,44,46)(30,41,37,33)(31,40,47,35);(3,11)(6,9)(10,16)(13,15)(17,19,21,23)(18,26,27,24)(20,29,32,38)(22,30,34,37)(25,42,43,39)(28,35,44,40)(31,46,47,45)(33,36,41,48);(2,8)(6,9)(7,14)(13,15)(18,27)(24,26)(25,43)(28,44)(31,47)(35,40)(39,42)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531567);