#356516

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

1230 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)(7,15,19,17)(8,12)(9,13)(10,14)(11,18,25,24)(16,29,32,30)(20,26)(21,27)(22,28)(23,31,35,34)(33,36);(5,7)(6,11,14,25)(8,16)(9,18)(10,19)(12,23,27,35)(13,15,28,17)(20,31)(21,32)(22,24)(26,29,36,30)(33,34);(2,4)(5,8,9,20,10,21,22,33)(6,12,13,26,14,27,28,36)(7,16,18,31,19,32,24,34)(11,23,15,29,25,35,17,30);(5,6)(7,17,19,15)(8,12)(9,13)(10,14)(11,24,25,18)(16,30,32,29)(20,26)(21,27)(22,28)(23,34,35,31)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,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, 263712);