#7273901

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1166 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7181097);