#7504405

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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2255 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7411601);