#7297241

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,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, 7204437);