#7512241

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419437);