#356545

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,g^b=gi,h^d=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263741);