#501400

order	512
nilpotency class	4
isoclinism	isoclinic class of #501400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408596);