#7308400

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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)(3,6)(4,7)(5,8)(17,18,23,29)(19,31,35,25)(20,36,21,27)(22,30,24,28)(26,38,43,37)(32,41,33,46)(34,42,39,44)(40,48,45,47);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,32)(21,33)(22,39)(23,35)(24,34)(26,40)(27,41)(28,44)(29,31)(30,42)(36,46)(37,48)(38,47)(43,45);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,43)(19,32,35,33)(22,37,24,38)(25,40,31,45)(27,28,36,30)(34,47,39,48)(41,44,46,42);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,21,23,20)(18,27,29,36)(19,33,35,32)(22,37,24,38)(25,41,31,46)(26,28,43,30)(34,47,39,48)(40,44,45,42);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,34,35,39)(20,37,21,38)(25,42,31,44)(26,36,43,27)(32,47,33,48)(40,41,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=f,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,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, 7215596);