#380395

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 287591);