#6846376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1184 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)(35,47)(36,44)(37,46)(38,42)(39,52)(40,43)(41,48)(45,50)(49,56)(51,54)(53,55);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(27,35)(28,39)(30,36)(32,45)(33,48)(38,50)(40,41)(42,47)(43,52)(46,54)(49,53);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,36,50,29,38,30,35)(26,32,44,45,31,42,34,47)(28,37,39,53,40,51,41,49)(33,46,52,55,43,54,48,56);(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)(28,40)(32,42)(34,44)(37,51)(39,41)(45,47)(49,53);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,41,29,40,36,39)(26,33,34,48,31,43,44,52)(27,37,35,49,38,51,50,53)(32,46,47,56,42,54,45,55)

Polycyclic presentation:

2^9|a^2=fi,c^2=fh,d^2=fgi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6753572);