#502144

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24291	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409340);