#7572008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,25)(22,26)(23,28)(24,33)(29,36)(30,42)(32,40)(34,39)(35,38)(37,44)(41,43)(45,47)(46,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,34)(23,27)(24,38)(26,39)(29,45)(31,40)(33,35)(36,47)(41,48)(43,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,28)(18,23,25,27)(20,29,33,43)(22,30,34,44)(24,36,31,41)(26,37,39,42)(32,46,35,45)(38,48,40,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,29)(21,33)(22,35)(23,36)(25,31)(26,40)(27,41)(28,43)(30,45)(32,34)(37,47)(38,39)(42,48)(44,46);(2,6)(7,8)(10,14)(13,15)(18,25)(19,28)(20,33)(26,39)(30,44)(32,35)(36,41)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7479204);