#424320

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1499 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)(5,6,10,16,12,17,11,29)(7,9,15,26,19,27,28,13)(8,20,23,33,25,34,24,14)(18,30,32,22,21,31,35,36);(5,7,12,19)(6,13,17,26)(8,21,25,18)(9,16,27,29)(10,28,11,15)(14,31,33,30)(20,22,34,36)(23,32,24,35);(5,8)(6,14,17,33)(7,18,19,21)(9,22)(10,23)(11,24)(12,25)(13,30)(15,32,28,35)(16,20,29,34)(26,31)(27,36);(2,4)(5,9,10,26,12,27,11,13)(6,15,16,19,17,28,29,7)(8,22,23,31,25,36,24,30)(14,32,20,21,33,35,34,18)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=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, 331516);