#416781

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

714 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)(5,6,10,18,12,16,11,17)(7,19,22,32,23,13,29,34)(8,24,26,14,27,15,9,28)(20,35,25,31,21,33,36,30);(1,2,4,3)(5,7,11,22,12,23,10,29)(6,13,17,34,16,19,18,32)(8,25,9,21,27,36,26,20)(14,33,24,30,28,35,15,31);(1,3,4,2)(5,8,10,26,12,27,11,9)(6,14,17,24,16,28,18,15)(7,20,22,25,23,21,29,36)(13,30,32,33,19,31,34,35);(5,9,11,27,12,26,10,8)(6,15,18,28,16,24,17,14)(7,21,22,36,23,20,29,25)(13,31,32,35,19,30,34,33)

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

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, 323977);