#7512368

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

1559 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)(19,23)(20,31)(21,26)(22,25)(24,32)(27,35)(28,42)(29,38)(30,37)(33,45)(34,40)(36,43)(39,46)(41,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,25,37)(20,28,32,43)(21,27,34,29)(24,36,31,42)(26,35,40,38)(33,41,46,44)(39,47,45,48);(10,14)(13,15)(19,27)(21,34)(23,35)(26,40)(28,41)(29,30)(33,46)(36,47)(37,38)(39,45)(42,48)(43,44);(2,5)(6,8)(11,16)(13,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,31)(26,39)(27,41)(29,44)(30,43)(34,46)(35,47)(37,42)(38,48)(40,45);(1,3,4,7)(2,6,5,8)(18,25)(20,32)(23,37)(26,40)(28,43)(33,46)(35,38)(41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,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, 7419564);