#368549

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=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, 275745);