#6489711

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6396907);