#6497979

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

Polycyclic presentation:

2^9|a^2=fhi,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, 6405175);