#395989

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1300 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,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, 303185);