#362870

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1152 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)(5,6,10,15)(7,12)(8,21)(9,13)(11,14,23,33)(16,17,29,34)(18,32)(19,30)(20,31)(22,35,36,28)(24,26)(25,27);(5,7,11,20,10,18,23,19)(6,12,14,31,15,32,33,30)(8,22,13,29,25,36,24,16)(9,17,27,35,26,34,21,28);(2,4)(5,8,23,13,10,25,11,24)(6,9,14,21,15,26,33,27)(7,16,19,22,18,29,20,36)(12,28,31,34,32,35,30,17);(5,9,10,26)(6,13,15,24)(7,17,18,34)(8,14,25,33)(11,27,23,21)(12,29,32,16)(19,28,20,35)(22,31,36,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=i,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, 270066);