#6465597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6372793);