#438370

order	512
nilpotency class	3
isoclinism	isoclinic class of #422540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1153 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:

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

Polycyclic presentation:

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

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, 345566);