#6702677

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1198 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,30,34)(27,31,38,48)(28,39,41,32)(29,33,43,51)(35,45,37,47)(36,53,55,46)(40,56,54,49)(42,52,44,50);(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27)(26,31)(28,40)(29,35)(30,38)(32,49)(33,45)(34,48)(36,42)(37,43)(39,56)(41,54)(44,55)(46,50)(47,51)(52,53);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(27,35)(28,41)(29,43)(31,45)(32,39)(33,51)(36,54)(37,38)(40,55)(46,56)(47,48)(49,53);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,41)(32,39)(36,55)(40,54)(42,44)(46,53)(49,56)(50,52);(11,21)(15,18)(19,20)(23,24)(25,28,30,41)(26,32,34,39)(27,36,38,55)(29,44,43,42)(31,46,48,53)(33,52,51,50)(35,40,37,54)(45,49,47,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=fg,e^2=i,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, 6609873);