#6465678

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3074 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)(5,6,9,13,11,16,10,15)(7,17,18,28,21,27,20,12)(8,14,24,29,23,32,25,31)(19,33,34,36,35,30,22,26);(2,4)(5,7)(6,12)(8,22)(9,20)(10,18)(11,21)(13,27)(14,30)(15,17)(16,28)(19,25)(23,34)(24,35)(26,31)(29,36)(32,33);(6,13,16,15)(7,18,21,20)(8,23)(14,31,32,29)(19,22,35,34)(24,25)(26,36)(30,33);(5,8,10,25,11,23,9,24)(6,14,15,31,16,32,13,29)(7,19,20,22,21,35,18,34)(12,26,27,30,28,36,17,33);(5,9,11,10)(6,13,16,15)(7,20,21,18)(8,24,23,25)(12,27,28,17)(14,29,32,31)(19,22,35,34)(26,30,36,33)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,e^2=i,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,d^c=di

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, 6372874);