#7329279

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=fi,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=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, 7236475);