#7624321

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	#62266	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1240 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)(19,28)(20,26)(21,34)(22,32)(23,25)(24,30)(27,36)(29,45)(31,44)(33,42)(35,40)(37,47)(38,43)(39,46)(41,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,24)(20,32)(21,27)(22,26)(23,25)(28,30)(29,41)(31,39)(33,47)(34,36)(35,43)(37,42)(38,40)(44,46)(45,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,29)(21,35)(22,38)(23,31)(25,39)(26,40)(27,43)(28,34)(30,36)(32,41)(33,46)(37,48)(42,45)(44,47);(3,11)(6,9)(10,16)(13,15)(18,25)(24,39)(26,32)(27,42)(28,44)(34,47)(40,41)(43,45);(2,8)(6,9)(7,14)(13,15)(17,20,23,22)(18,26,25,32)(19,29,31,38)(21,33,37,36)(24,40,39,41)(27,34,42,47)(28,45,44,43)(30,35,46,48)

Polycyclic presentation:

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

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, 7531517);