#365773

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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)(7,10)(8,19,24,17)(9,25)(11,15)(12,16)(13,23,20,22)(14,26)(18,28)(21,33,36,29)(27,30)(31,34,32,35);(1,2)(3,4)(5,7,11,16,12,15,10,6)(8,20,22,17,24,13,23,19)(9,18,27,30,28,25,26,14)(21,33,34,31,36,29,35,32);(2,4)(5,8)(6,13,16,20)(7,17,15,19)(9,21)(10,22)(11,23)(12,24)(14,29,30,33)(18,31,25,32)(26,34)(27,35)(28,36);(5,9,11,27,12,28,10,26)(6,14,7,18,16,30,15,25)(8,21,23,35,24,36,22,34)(13,29,17,31,20,33,19,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 272969);