#7623502

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1315 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,31)(21,26)(22,36)(23,24)(25,34)(27,38)(29,45)(30,43)(32,44)(33,47)(35,40)(37,41)(39,48)(42,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,23)(18,24,26,28)(20,32,33,29)(22,30,35,38)(25,41,42,39)(27,40,43,36)(31,45,47,44)(34,48,46,37);(3,10)(6,9)(11,16)(13,15)(17,20,21,33)(18,25,26,42)(19,29,23,32)(22,37,35,48)(24,39,28,41)(27,44,43,45)(30,46,38,34)(31,40,47,36);(2,7)(6,9)(8,14)(13,15)(18,26)(24,28)(25,42)(27,43)(31,47)(36,40)(39,41)(44,45);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530698);