#7369257

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

1390 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)(17,18,22,28,23,29,21,27)(19,24,31,40,32,41,36,43)(20,33,34,42,35,26,37,25)(30,44,46,48,47,39,45,38);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,29,41)(20,30,35,47)(21,36,22,31)(25,38,42,48)(26,39,33,44)(27,43,28,40)(34,46,37,45);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,28)(27,33)(29,42)(31,45)(32,47)(36,46)(39,40)(41,48)(43,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,26)(19,30)(21,37)(22,34)(23,35)(24,39)(25,28)(27,42)(29,33)(31,46)(32,47)(36,45)(38,40)(41,44)(43,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,40,41,43)(25,33,42,26)(30,45,47,46)(38,39,48,44)

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,d^a=di,e^a=ef,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, 7276453);