#6497978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fgi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405174);