#416813

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 324009);