#380190

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
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

984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287386);