#7172048

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1247 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)(10,13)(11,15)(17,18,22,29,23,28,38,27)(19,30,32,42,33,24,39,43)(20,34,36,44,21,26,35,25)(31,45,48,40,37,41,47,46);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19)(18,24)(20,31)(21,37)(22,32)(23,33)(25,40)(26,45)(27,42)(28,30)(29,43)(34,41)(35,47)(36,48)(38,39)(44,46);(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,25,28,44)(19,31,33,37)(22,35,38,36)(24,40,30,46)(26,27,34,29)(32,47,39,48)(41,43,45,42);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,21,23,20)(18,26,28,34)(19,31,33,37)(22,35,38,36)(24,41,30,45)(25,27,44,29)(32,48,39,47)(40,43,46,42);(9,11,12,15)(10,14,13,16)(17,22,23,38)(18,27,28,29)(19,32,33,39)(20,35,21,36)(24,42,30,43)(25,34,44,26)(31,47,37,48)(40,41,46,45)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,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, 7079244);