#7339721

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fgi,d^2=f,e^2=fi,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^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7246917);