#416782

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323978);