#7278175

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#31060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7185371);