#6529250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	#35191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2854 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)(23,26)(24,32)(25,29)(27,31)(28,34)(30,35)(33,36);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,22)(24,28,33,30)(25,29,27,31)(32,35,36,34);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,33)(22,28,23,30)(25,34,27,35)(29,36,31,32);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,23)(24,33)(29,31)(34,35);(6,12)(10,13)(11,20)(18,19)(21,25,26,27)(22,29,23,31)(24,34,33,35)(28,36,30,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6436446);