#7326721

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(17,18,22,27)(19,23,30,26)(20,31,34,37)(21,25,36,42)(24,32,40,28)(29,38,35,41)(33,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,35,36,29)(24,31,40,37)(25,41,42,38)(33,43,46,45)(39,47,48,44);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,37,27,40,26,31)(21,33,35,45,36,46,29,43)(25,39,38,47,42,48,41,44);(3,8)(5,7)(10,14)(11,16)(17,21)(18,25)(19,29)(20,33)(22,36)(23,38)(24,39)(26,41)(27,42)(28,43)(30,35)(31,44)(32,45)(34,46)(37,47)(40,48);(2,6)(5,7)(10,14)(13,15)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,35,36,29)(24,31,40,37)(25,41,42,38)(33,43,46,45)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,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, 7233917);