#412534

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

958 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)(5,6,9,13,10,15,22,14)(7,11,18,16,20,27,19,26)(8,21,23,12,24,28,35,29)(17,31,32,25,34,30,33,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,18)(10,20)(12,25)(13,16)(14,26)(15,27)(19,22)(21,31)(23,32)(24,34)(28,30)(29,36)(33,35);(2,3)(5,6)(7,16)(8,12)(9,14)(10,15)(11,18)(13,22)(17,30)(19,27)(20,26)(21,23)(24,29)(25,32)(28,35)(31,34)(33,36);(5,8,22,35,10,24,9,23)(6,12,13,28,15,29,14,21)(7,17,19,33,20,34,18,32)(11,25,16,30,27,36,26,31)

Polycyclic presentation:

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

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, 319730);