#6481685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	C₂×#3219	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2899 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,33)(24,38)(25,32)(26,31)(27,30)(28,36)(29,41)(34,46)(35,45)(37,44)(39,43)(40,47)(42,48)(49,51)(50,52);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,39)(25,37)(26,36)(27,35)(29,46)(30,44)(31,45)(32,33)(34,42)(38,49)(40,50)(41,51)(43,47)(48,52);(7,13)(10,17)(14,15)(18,19)(21,24,27,42)(22,29,30,47)(23,34,35,39)(25,41,26,40)(28,43,44,46)(31,38,32,48)(33,49,45,52)(36,51,37,50);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,44)(33,45)(34,39)(36,37)(43,46)(49,52)(50,51);(6,12)(10,17)(11,20)(18,19)(22,30)(23,35)(29,47)(31,32)(34,39)(36,37)(38,48)(50,51)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,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=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388881);