#356517

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	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(5,6,9,14,11,15,10,25)(7,16,20,12,21,18,19,27)(8,13,22,28,24,29,23,35)(17,30,33,26,34,31,32,36);(5,7,10,20,11,21,9,19)(6,12,14,16,15,27,25,18)(8,17,23,33,24,34,22,32)(13,26,28,30,29,36,35,31);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,28,15,29,25,35)(7,17,19,32,21,34,20,33)(12,26,16,30,27,36,18,31);(5,6,9,14,11,15,10,25)(7,18,20,27,21,16,19,12)(8,13,22,28,24,29,23,35)(17,31,33,36,34,30,32,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,g^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, 263713);