#371871

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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)(5,6,12,16,11,17,10,21)(7,8,14,23,20,24,19,13)(9,25,28,31,27,35,26,15)(18,32,30,36,34,29,33,22);(2,4)(5,7)(6,13,17,23)(8,21,24,16)(9,18)(10,19)(11,20)(12,14)(15,29,31,32)(22,35,36,25)(26,33)(27,34)(28,30);(2,4)(5,8,12,13,11,24,10,23)(6,14,16,7,17,19,21,20)(9,22,28,29,27,36,26,32)(15,30,25,18,31,33,35,34);(2,4)(5,9,10,26,11,27,12,28)(6,15,16,25,17,31,21,35)(7,18,19,33,20,34,14,30)(8,22,23,32,24,36,13,29)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 279067);