#7020454

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6927650);