#459926

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367122);