#7368883

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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)(3,7)(4,6)(5,8)(17,18,23,37,22,27,21,26)(19,28,30,24,32,39,31,40)(20,33,35,42,34,25,36,41)(29,44,47,48,45,43,46,38);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(20,34)(25,33)(29,45)(35,36)(38,48)(41,42)(43,44)(46,47);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,38)(26,39)(27,40)(28,37)(33,48)(34,45)(35,46)(36,47)(41,43)(42,44);(3,8)(5,7)(10,14)(11,16)(17,20,22,34)(18,25,27,33)(19,29,32,45)(21,36,23,35)(24,38,40,48)(26,42,37,41)(28,43,39,44)(30,47,31,46);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,39,40,28)(25,41,33,42)(29,46,45,47)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7276079);