#6890112

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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)(25,26,32,37)(27,33,40,51)(28,41,29,35)(30,36,31,46)(34,44,52,42)(38,48,43,53)(39,49,45,50)(47,54,56,55);(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)(26,33)(28,38)(29,43)(30,45)(31,39)(32,40)(34,47)(35,53)(36,50)(37,51)(41,48)(42,55)(44,54)(46,49)(52,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,32,29)(26,34,37,52)(27,38,40,43)(30,42,31,44)(33,47,51,56)(35,46,41,36)(39,54,45,55)(48,50,53,49);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,29,32,28)(26,35,37,41)(27,38,40,43)(30,42,31,44)(33,48,51,53)(34,46,52,36)(39,55,45,54)(47,50,56,49);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,42,29,44)(33,49,51,50)(34,35,52,41)(38,54,43,55)(47,48,56,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,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, 6797308);