#6971822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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

1320 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)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,30,35)(27,31,38,34)(28,39,41,44)(29,33,43,50)(32,46,48,36)(37,45,42,49)(40,52,54,55)(47,53,56,51);(2,5)(3,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,30,38)(26,31,35,34)(28,36,41,46)(29,37,43,42)(32,44,48,39)(33,45,50,49)(40,51,54,53)(47,55,56,52);(1,2)(3,6)(4,5)(7,8)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,38,46,30,41,27,36)(26,32,31,44,35,48,34,39)(29,40,42,53,43,54,37,51)(33,47,45,55,50,56,49,52);(2,5)(6,8)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29)(26,33)(27,37)(28,40)(30,43)(31,45)(32,47)(34,49)(35,50)(36,51)(38,42)(39,52)(41,54)(44,55)(46,53)(48,56);(1,3,4,7)(2,6,5,8)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(25,27,30,38)(26,31,35,34)(28,36,41,46)(29,42,43,37)(32,44,48,39)(33,49,50,45)(40,53,54,51)(47,52,56,55)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6879018);