#388373

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

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

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, 295569);