#356240

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263436);