#6529695

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	#37280	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6436891);