#7301393

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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

1637 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)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,26,36,44)(25,31,41,28)(29,39,35,43)(32,47,34,48)(40,46,42,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,36,35)(25,30,41,38)(26,39,44,43)(32,46,34,45)(40,47,42,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,34,36,32)(24,38,37,30)(26,42,44,40)(29,46,35,45)(39,47,43,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,29)(20,32)(22,35)(23,36)(24,39)(25,40)(27,44)(28,45)(30,47)(31,46)(33,34)(37,43)(38,48)(41,42);(2,6)(5,7)(10,14)(13,15)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,35,36,29)(25,30,41,38)(26,43,44,39)(32,45,34,46)(40,48,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7208589);