#355559

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 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)(6,12)(8,21)(9,10)(13,26)(14,29)(15,16)(17,28)(18,32)(19,24)(22,23)(31,35)(33,34);(5,6,10,16,11,15,9,12)(7,17,19,28,20,26,24,13)(8,14,22,30,23,29,21,25)(18,31,32,36,33,35,34,27);(2,4)(5,7)(6,13,15,28)(8,18)(9,19)(10,24)(11,20)(12,17,16,26)(14,27,29,36)(21,32)(22,34)(23,33)(25,31,30,35);(5,8,10,22,11,23,9,21)(6,14,16,30,15,29,12,25)(7,18,24,34,20,33,19,32)(13,27,26,35,28,36,17,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262755);