#6745056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1192 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)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,36)(28,41)(29,44)(30,33)(31,35)(32,39)(34,43)(37,46)(38,47)(40,50)(42,48)(45,49)(51,53)(52,56)(54,55);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,38)(29,40)(30,39)(31,37)(33,36)(34,47)(35,50)(41,51)(42,52)(43,53)(44,46)(45,54)(48,55)(49,56);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(29,31)(32,46)(34,41)(36,50)(38,52)(39,40)(42,45)(47,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,47)(33,41)(35,49)(36,51)(37,52)(39,53)(40,54)(44,48)(46,55)(50,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29)(26,35)(27,39)(28,42)(30,31)(33,44)(34,48)(41,49)(43,45)(46,50)(47,51)(52,54)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652252);