#7369129

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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)(3,6)(4,5)(7,8)(17,18,22,27,23,28,21,34)(19,24,30,40,31,41,35,39)(20,26,32,43,33,42,36,25)(29,38,45,47,46,48,44,37);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,41)(20,29,33,46)(21,35,22,30)(25,37,43,47)(26,38,42,48)(27,40,34,39)(32,45,36,44);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,37)(26,34)(27,42)(28,43)(30,44)(31,46)(35,45)(38,39)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,26)(19,29)(21,36)(22,32)(23,33)(24,38)(25,34)(27,43)(28,42)(30,45)(31,46)(35,44)(37,39)(40,47)(41,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,27,28,34)(19,30,31,35)(20,32,33,36)(24,39,41,40)(25,42,43,26)(29,44,46,45)(37,38,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,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, 7276325);