#6797350

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(25,26,29,33)(27,35,37,31)(28,39,41,55)(30,43,40,34)(32,46,47,42)(36,50,52,54)(38,45,51,49)(44,53,48,56);(3,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,29,37)(26,31,33,35)(28,36,41,52)(30,38,40,51)(32,44,47,48)(34,45,43,49)(39,54,55,50)(42,53,46,56);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,40,46,29,41,30,42)(26,32,43,55,33,47,34,39)(27,36,51,56,37,52,38,53)(31,44,49,50,35,48,45,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,33)(27,37)(28,41)(34,43)(38,51)(42,46)(44,48)(50,54);(1,4,5,3)(2,6,7,8)(11,21)(15,18)(19,20)(23,24)(27,37)(28,41)(31,35)(32,47)(38,51)(39,55)(42,46)(45,49)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=h,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^c=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, 6704546);