#367088

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×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,d^2=e,e^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, 274284);