#365075

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 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)(5,6,12,15,11,16,10,28)(7,17,21,8,20,23,19,13)(9,24,27,30,26,36,25,14)(18,22,34,35,33,29,32,31);(5,7)(6,13,16,8)(9,18)(10,19)(11,20)(12,21)(14,29,30,22)(15,17,28,23)(24,31,36,35)(25,32)(26,33)(27,34);(2,4)(5,8)(6,7)(9,22)(10,23)(11,13)(12,17)(14,18)(15,19)(16,20)(21,28)(24,32)(25,35)(26,29)(27,31)(30,33)(34,36);(5,9,12,27,11,26,10,25)(6,14,28,36,16,30,15,24)(7,18,21,34,20,33,19,32)(8,22,17,31,13,29,23,35)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=ei,e^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,g^b=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, 272271);