#356546

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

919 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)(7,18,22,33)(8,24)(9,15)(10,16)(11,17)(12,29)(13,21,31,23)(14,26)(19,34,30,27)(20,32,36,35)(25,28);(5,7,12,23,11,22,10,21)(6,13,16,18,17,31,29,33)(8,19,15,32,26,30,25,35)(9,27,24,36,28,34,14,20);(2,4)(5,8,10,25,11,26,12,15)(6,14,16,9,17,24,29,28)(7,19,21,35,22,30,23,32)(13,20,18,27,31,36,33,34);(5,9,10,24,11,28,12,14)(6,15,16,8,17,25,29,26)(7,20,23,34,22,36,21,27)(13,30,33,35,31,19,18,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=eg,e^2=i,h^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, 263742);