#399127

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

1241 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:

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

Polycyclic presentation:

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

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, 306323);