#7339793

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(19,27)(20,25)(21,23)(22,34)(24,30)(26,33)(28,40)(29,39)(31,36)(32,45)(35,42)(37,43)(38,44)(41,47)(46,48);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(18,23)(19,28)(25,36)(26,34)(29,41)(30,42)(38,45)(43,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19,22,30,21,28,33,42)(18,24,26,27,23,35,34,40)(20,29,32,43,31,41,44,48)(25,37,38,39,36,46,45,47);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,20,21,31)(18,25,23,36)(19,29,28,41)(22,32,33,44)(24,37,35,46)(26,38,34,45)(27,39,40,47)(30,43,42,48);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(18,23)(19,28)(20,31)(26,34)(30,42)(32,44)(37,46)(39,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,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^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, 7246989);