#7504367

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2334 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)(19,30)(20,25)(21,27)(22,28)(23,29)(24,34)(26,38)(31,47)(32,41)(33,43)(35,42)(36,45)(37,44)(39,46)(40,48);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19,22,34)(18,24,28,30)(20,36,38,32)(21,35,23,33)(25,41,26,45)(27,43,29,42)(31,39,48,37)(40,46,47,44);(10,14)(13,16)(17,20)(18,25)(19,31)(21,37)(22,38)(23,39)(24,40)(26,28)(27,44)(29,46)(30,47)(32,35)(33,36)(34,48)(41,42)(43,45);(2,6)(3,8)(10,14)(13,16)(17,20)(18,26)(19,32)(21,39)(22,38)(23,37)(24,41)(25,28)(27,44)(29,46)(30,45)(31,35)(33,48)(34,36)(40,42)(43,47);(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,37,38,39)(24,42,30,43)(25,44,26,46)(31,36,48,32)(40,41,47,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,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, 7411563);