#6704314

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1219 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,33)(27,31,37,47)(28,39,41,32)(30,34,43,51)(35,53,54,45)(36,46,38,48)(40,56,55,49)(42,52,44,50);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,40)(29,37)(30,36)(32,49)(33,47)(34,46)(35,42)(38,43)(39,56)(41,55)(44,54)(45,50)(48,51)(52,53);(2,5)(6,8)(25,28,29,41)(26,32,33,39)(27,35,37,54)(30,44,43,42)(31,45,47,53)(34,52,51,50)(36,40,38,55)(46,49,48,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(49,56)(50,52);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,36)(28,41)(30,43)(31,46)(32,39)(34,51)(35,55)(37,38)(40,54)(45,56)(47,48)(49,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=gi,e^2=f,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,h^c=hi,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, 6611510);