#6875557

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6782753);