#7339529

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(19,28,29,40)(20,25,32,41)(21,26,22,37)(24,35,38,31)(30,45,33,48)(34,43,36,44)(39,46,42,47);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(19,29)(20,32)(24,38)(25,41)(28,40)(31,35)(34,36)(43,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,35,23,29,22,31)(18,24,37,40,27,38,26,28)(20,33,36,47,32,30,34,46)(25,42,43,45,41,39,44,48);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,20)(18,25)(19,30)(21,36)(22,34)(23,32)(24,39)(26,44)(27,41)(28,45)(29,33)(31,47)(35,46)(37,43)(38,42)(40,48);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(17,21,23,22)(18,26,27,37)(19,31,29,35)(20,34,32,36)(24,40,38,28)(25,43,41,44)(30,46,33,47)(39,45,42,48)

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=dh,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, 7246725);