#7126781

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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)(9,10,14,11)(12,15,13,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,33)(28,35)(29,34)(30,37)(31,39)(32,38)(36,43)(40,44)(41,46)(42,45)(47,48);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,22,30)(18,23,27,37)(20,28,31,41)(21,29,33,43)(24,34,26,36)(25,35,39,46)(32,42,44,48)(38,45,40,47);(9,11,14,10)(12,15,13,16)(18,24)(21,33)(23,34)(25,38)(26,27)(29,43)(32,44)(35,45)(36,37)(39,40)(42,48)(46,47);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,22,31)(18,25,27,39)(19,28,30,41)(21,32,33,44)(23,35,37,46)(24,38,26,40)(29,42,43,48)(34,45,36,47);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(20,31)(25,39)(28,41)(32,44)(35,46)(38,40)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=f,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,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, 7033977);