#6465686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3150 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6372882);