#367092

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efghi,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, 274288);