#354122

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2810 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)(7,12,19,27)(8,21)(9,14)(10,16)(11,15)(13,22)(17,31,34,35)(18,26,20,30)(23,28)(24,29)(25,32,36,33);(5,7,10,20,11,19,9,18)(6,12,14,26,15,27,16,30)(8,17,24,33,23,34,22,32)(13,25,28,31,29,36,21,35);(1,2,3,4)(5,8,9,22,11,23,10,24)(6,13,16,21,15,29,14,28)(7,17,18,32,19,34,20,33)(12,25,30,35,27,36,26,31);(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,27,30)(13,28,29,21)(17,32,34,33)(25,31,36,35)

Polycyclic presentation:

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

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, 261318);