#6846663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1186 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)(27,32)(28,33)(29,31)(30,34)(35,48)(36,45)(37,43)(38,47)(39,53)(40,50)(41,44)(42,49)(46,51)(52,56)(54,55);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(27,35)(28,39)(30,36)(32,46)(33,49)(37,51)(41,42)(43,48)(44,53)(47,50)(52,54);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,36,51,29,37,30,35)(26,32,45,46,31,43,34,48)(28,40,39,52,41,38,42,54)(33,50,53,56,44,47,49,55);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(26,31)(27,37)(33,44)(34,45)(35,51)(38,40)(49,53)(52,54);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,36,39,29,41,30,42)(26,33,45,53,31,44,34,49)(27,38,51,54,37,40,35,52)(32,47,46,55,43,50,48,56)

Polycyclic presentation:

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

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, 6753859);