#7207861

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207850
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

1433 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,38)(20,30,33,40)(21,34,37,26)(25,28,42,31)(29,43,35,39)(32,48,36,47)(41,46,44,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,38)(20,31,33,28)(21,35,37,29)(25,40,42,30)(26,43,34,39)(32,45,36,46)(41,47,44,48);(11,15)(13,16)(17,20,19,28,23,33,22,31)(18,25,38,40,27,42,24,30)(21,36,29,46,37,32,35,45)(26,44,43,47,34,41,39,48);(3,8)(5,7)(10,14)(11,15)(17,21,23,37)(18,26,27,34)(19,29,22,35)(20,32,33,36)(24,39,38,43)(25,41,42,44)(28,45,31,46)(30,47,40,48);(2,6)(5,7)(11,15)(13,16)(21,37)(26,34)(29,35)(32,36)(39,43)(41,44)(45,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7115057);