#7532094

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,41)(20,25,35,45)(21,36,22,26)(28,38,32,43)(29,40,37,39)(31,42,47,48)(33,46,34,44);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,31)(21,29)(22,37)(23,30)(25,42)(26,39)(27,41)(28,34)(32,33)(35,47)(36,40)(38,44)(43,46)(45,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,24)(20,32)(21,37)(22,29)(23,30)(25,43)(26,40)(27,41)(28,35)(31,33)(34,47)(36,39)(38,45)(42,46)(44,48);(10,14)(13,15)(17,20,23,35)(18,25,27,45)(19,28,30,32)(21,34,22,33)(24,38,41,43)(26,46,36,44)(29,47,37,31)(39,42,40,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,37)(20,33,35,34)(24,39,41,40)(25,44,45,46)(28,31,32,47)(38,48,43,42)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7439290);