#7037433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1242 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6944629);