#367384

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274580);