#428868

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

927 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336064);