#7339204

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,34)(27,35,36,48)(28,32,39,46)(29,41,42,33)(31,43,44,38)(37,49,50,56)(40,53,52,47)(45,54,55,51);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(27,36)(28,39)(31,44)(32,46)(35,48)(38,43)(40,52)(47,53);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,38,30,36,42,43)(26,31,33,35,34,44,41,48)(28,37,40,51,39,50,52,54)(32,45,47,49,46,55,53,56);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28)(26,32)(27,37)(29,40)(30,39)(31,45)(33,47)(34,46)(35,49)(36,50)(38,51)(41,53)(42,52)(43,54)(44,55)(48,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,39)(32,46)(37,50)(40,52)(45,55)(47,53)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,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=dfh,e^b=ehi,d^c=dhi,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, 7246400);