#445741

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

827 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)(5,6,9,13,11,16,10,15)(7,17,21,26,19,12,20,28)(8,22,25,30,23,29,24,14)(18,32,33,31,35,36,34,27);(5,7,9,21,11,19,10,20)(6,12,15,26,16,17,13,28)(8,18,25,33,23,35,24,34)(14,27,29,36,30,31,22,32);(6,13,16,15)(12,26,17,28)(14,29,30,22)(27,32,31,36);(2,4)(5,8,10,25,11,23,9,24)(6,14,13,29,16,30,15,22)(7,18,21,34,19,35,20,33)(12,27,26,32,17,31,28,36)

Polycyclic presentation:

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

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, 352937);