#6529217

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499761
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3056 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)(23,29)(24,32)(25,30)(26,27)(28,31)(33,35)(34,36);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,26)(22,27,30,29)(24,31,33,34)(28,32,36,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,35)(30,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23,25,26)(22,29,30,27)(24,31,33,34)(28,35,36,32);(6,12)(10,13)(11,20)(18,19)(21,23,25,26)(22,29,30,27)(24,31,33,34)(28,35,36,32)

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,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436413);