#6986626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,32,38)(27,33,42,51)(28,43,29,35)(30,37,31,36)(34,46,52,44)(39,48,45,53)(40,50,41,49)(47,55,56,54);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27)(26,33)(28,39)(29,45)(30,40)(31,41)(32,42)(34,47)(35,53)(36,49)(37,50)(38,51)(43,48)(44,54)(46,55)(52,56);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,28,32,29)(26,34,38,52)(27,39,42,45)(30,44,31,46)(33,47,51,56)(35,36,43,37)(40,54,41,55)(48,50,53,49);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,32,28)(26,35,38,43)(27,39,42,45)(30,44,31,46)(33,48,51,53)(34,36,52,37)(40,55,41,54)(47,50,56,49);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,32,31)(26,36,38,37)(27,40,42,41)(28,44,29,46)(33,49,51,50)(34,43,52,35)(39,54,45,55)(47,48,56,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6893822);