#6817317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgh,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi

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, 6724513);