#7204209

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#53151	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,28)(20,32)(21,26)(22,36)(23,27)(24,31)(25,33)(29,40)(30,45)(34,42)(35,41)(37,38)(39,46)(43,44)(47,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(19,29)(20,33)(24,38)(25,32)(28,40)(31,37)(34,35)(41,42);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,29)(25,39)(26,40)(27,38)(28,36)(32,48)(33,44)(34,46)(35,47)(41,43)(42,45);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,20,23,33)(18,25,27,32)(19,30,29,44)(21,35,22,34)(24,39,38,48)(26,42,36,41)(28,43,40,45)(31,47,37,46);(3,7)(5,8)(10,15)(14,16)(17,21,23,22)(18,26,27,36)(19,31,29,37)(20,34,33,35)(24,40,38,28)(25,41,32,42)(30,46,44,47)(39,43,48,45)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7111405);