#7278063

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	#31737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(17,18,23,28)(19,24,33,42)(20,34,36,26)(21,27,22,38)(25,37,43,35)(29,40,30,44)(31,41,32,47)(39,46,48,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,33)(25,39)(26,44)(27,41)(28,42)(30,36)(34,40)(35,45)(37,46)(38,47)(43,48);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,35)(22,37)(23,36)(24,39)(26,27)(28,43)(30,33)(31,45)(32,46)(34,38)(40,47)(41,44)(42,48);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,37)(22,35)(23,36)(24,40)(25,27)(28,34)(29,33)(31,45)(32,46)(38,43)(39,47)(41,48)(42,44);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,23,22)(18,27,28,38)(19,31,33,32)(20,35,36,37)(24,41,42,47)(25,26,43,34)(29,45,30,46)(39,44,48,40)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=h,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=dfi,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, 7185259);