#374358

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1754 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 281554);