#6529185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#35522	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(22,27)(23,29)(25,33)(28,34)(30,35)(32,36);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,23)(24,28,31,30)(25,29,33,27)(32,35,36,34);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,22)(24,30,31,28)(25,27,33,29)(32,34,36,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,26,31)(22,28,23,30)(25,32,33,36)(27,34,29,35);(6,12)(10,13)(11,20)(18,19)(24,31)(28,30)(32,36)(34,35)

Polycyclic presentation:

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

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, 6436381);