#424290

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1507 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)(5,6,9,14,11,15,10,25)(7,12,18,28,19,16,20,31)(8,21,23,13,24,22,26,29)(17,32,33,27,34,36,35,30);(5,6)(7,16)(8,22)(9,25)(10,14)(11,15)(12,19)(13,23)(17,32)(18,28)(20,31)(21,24)(26,29)(27,35)(30,33)(34,36);(1,2,3,4)(5,7,9,18,11,19,10,20)(6,12,14,28,15,16,25,31)(8,17,23,33,24,34,26,35)(13,27,22,36,29,30,21,32);(5,8,9,23,11,24,10,26)(6,13,14,22,15,29,25,21)(7,17,18,33,19,34,20,35)(12,27,28,36,16,30,31,32)

Polycyclic presentation:

2^9|a^2=e,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 331486);