#424341

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	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 331537);