#7238208

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,31,36)(27,32,40,47)(28,34,42,50)(29,35,30,33)(37,45,52,56)(38,46,39,44)(41,48,43,49)(51,54,53,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,31,40)(26,32,36,47)(28,37,42,52)(29,38,30,39)(33,44,35,46)(34,45,50,56)(41,51,43,53)(48,54,49,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(29,30)(32,44)(34,48)(35,36)(38,39)(41,43)(45,54)(46,47)(49,50)(51,53)(55,56);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,45)(33,48)(35,49)(36,50)(38,53)(39,51)(40,52)(44,54)(46,55)(47,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,35,36,33)(27,38,40,39)(28,41,42,43)(32,46,47,44)(34,48,50,49)(37,51,52,53)(45,54,56,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,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, 7145404);