#7510108

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,30,23)(20,32,34,24)(22,26,36,42)(28,39,31,37)(29,38,33,40)(35,45,46,41)(43,47,44,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,33)(21,30)(22,28)(24,40)(25,27)(26,37)(29,34)(31,36)(32,38)(35,44)(39,42)(41,48)(43,46)(45,47);(9,12,13,11)(10,14,15,16)(19,28)(22,36)(23,37)(26,42)(27,39)(29,43)(30,31)(33,44)(35,46)(38,47)(40,48)(41,45);(2,7)(3,8)(10,15)(14,16)(17,20)(18,24)(19,29)(21,34)(22,35)(23,38)(25,32)(26,41)(27,40)(28,43)(30,33)(31,44)(36,46)(37,47)(39,48)(42,45);(1,4,5,6)(2,3,7,8)(19,30)(20,34)(23,27)(24,32)(28,31)(35,46)(37,39)(41,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=fi,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, 7417304);