#7013258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(17,18,24,28,23,30,22,29)(19,25,38,41,33,43,32,42)(20,27,35,26,21,37,36,45)(31,46,47,39,34,40,48,44);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,34)(21,31)(22,32)(23,33)(24,38)(26,44)(27,40)(28,41)(29,42)(30,43)(35,48)(36,47)(37,46)(39,45);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,35)(24,36)(25,39)(27,28)(29,37)(30,45)(32,47)(33,34)(38,48)(40,42)(41,46)(43,44);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,27,30,37)(19,31,33,34)(22,35,24,36)(25,40,43,46)(26,29,45,28)(32,48,38,47)(39,41,44,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,32,33,38)(20,35,21,36)(25,41,43,42)(26,27,45,37)(31,47,34,48)(39,46,44,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6920454);