#7038322

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(17,18,22,27)(19,29,31,24)(20,33,21,26)(23,38,37,28)(25,36,42,34)(30,43,35,40)(32,41,47,45)(39,46,44,48);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,39)(26,43)(27,29)(28,41)(33,40)(34,46)(36,48)(37,47)(38,45)(42,44);(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,25,27,42)(19,30,31,35)(23,36,37,34)(24,39,29,44)(26,38,33,28)(32,48,47,46)(40,41,43,45);(3,8)(6,7)(10,13)(14,16)(17,21,22,20)(18,26,27,33)(19,30,31,35)(23,36,37,34)(24,40,29,43)(25,38,42,28)(32,46,47,48)(39,41,44,45);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(20,21)(25,42)(26,33)(30,35)(34,36)(39,44)(40,43)(46,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=i,e^2=fi,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,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, 6945518);