#6797365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(3,8)(4,6)(5,7)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,35)(27,36,31,32)(28,39,42,47)(29,43,46,34)(33,37,50,40)(38,48,44,51)(41,56,45,55)(49,54,52,53);(3,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,44,46,38)(33,39,50,47)(34,51,43,48)(41,53,45,54)(49,56,52,55);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,37,30,42,31,40)(26,33,32,47,35,50,36,39)(29,45,38,54,46,41,44,53)(34,52,48,55,43,49,51,56);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29)(26,34)(27,38)(28,41)(30,46)(31,44)(32,48)(33,49)(35,43)(36,51)(37,53)(39,55)(40,54)(42,45)(47,56)(50,52);(1,4,5,3)(2,6,7,8)(11,21)(15,18)(19,20)(23,24)(29,46)(34,43)(38,44)(41,45)(48,51)(49,52)(53,54)(55,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6704561);