#7005004

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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,e^2=i,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, 6912200);