#356502

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

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

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, 263698);