#384330

order	512
nilpotency class	3
isoclinism	isoclinic class of #219696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 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)(7,18,23,34)(8,24)(9,27,29,15)(10,16)(11,30)(12,17)(13,21,32,22)(14,20)(19,35,28,31)(25,33)(26,36);(1,2)(3,4)(5,7)(6,13)(8,9)(10,22)(11,21)(12,23)(14,27)(15,33)(16,34)(17,32)(18,30)(19,20)(24,31)(25,28)(26,29)(35,36);(2,4)(5,8,11,20,12,26,10,25)(6,14,16,24,17,33,30,36)(7,19,22,9,23,28,21,29)(13,31,18,27,32,35,34,15);(5,9,10,19,12,29,11,28)(6,15,16,31,17,27,30,35)(7,20,21,8,23,25,22,26)(13,24,18,33,32,36,34,14)

Polycyclic presentation:

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

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, 291526);