#6993750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fhi,d^2=fh,e^2=fh,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

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, 6900946);