#7505027

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=f,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, 7412223);