#6993764

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fh,d^2=fh,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

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, 6900960);