#7512433

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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)(19,24)(20,32)(21,35)(22,26)(23,27)(25,34)(28,37)(29,44)(30,39)(31,40)(33,41)(36,42)(38,46)(43,48)(45,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,27,40)(20,29,34,46)(21,30,22,28)(25,38,32,44)(26,37,35,39)(33,43,36,45)(41,48,42,47);(10,14)(13,15)(19,28)(21,22)(24,37)(26,35)(29,43)(30,31)(33,36)(38,47)(39,40)(41,42)(44,48)(45,46);(2,5)(6,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,42)(27,32)(28,43)(30,45)(31,46)(35,41)(37,47)(39,48)(40,44);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,35)(19,30,31,28)(20,33,34,36)(24,37,40,39)(25,41,32,42)(29,43,46,45)(38,48,44,47)

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,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7419629);