#7504369

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2304 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7411565);