#384428

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

943 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291624);