#7297243

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,34)(27,36,38,31)(28,33,41,52)(30,44,43,35)(32,45,50,42)(37,51,40,47)(39,48,55,53)(46,54,49,56);(2,5)(3,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,55)(32,49,50,46)(33,51,52,47)(35,48,44,53)(42,56,45,54);(1,2)(3,6)(4,5)(7,8)(25,28,29,41)(26,32,34,50)(27,37,38,40)(30,45,43,42)(31,46,36,49)(33,35,52,44)(39,56,55,54)(47,48,51,53);(2,5)(6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,47)(32,44)(34,52)(35,50)(36,51)(38,40)(39,54)(43,45)(46,53)(48,49)(55,56);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,24)(22,23)(26,34)(28,41)(31,36)(35,44)(37,40)(42,45)(48,53)(54,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 7204439);