#6889918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6797114);