#7512050

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419246);