#395736

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1821 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)(5,6,12,15,11,17,10,16)(7,18,14,25,21,26,20,9)(8,13,27,30,24,32,23,31)(19,33,35,22,34,36,29,28);(5,7,10,20,11,21,12,14)(6,9,15,18,17,25,16,26)(8,19,23,29,24,34,27,35)(13,28,30,33,32,22,31,36);(2,4)(5,8,11,24)(6,13,17,32)(7,19,21,34)(9,22,25,28)(10,27,12,23)(14,29,20,35)(15,31,16,30)(18,33,26,36);(5,9)(6,14)(7,15)(8,22)(10,18)(11,25)(12,26)(13,29)(16,21)(17,20)(19,31)(23,36)(24,28)(27,33)(30,34)(32,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi,h^d=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, 302932);