#424339

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	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 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)(5,6,9,15,11,16,10,12)(7,17,20,29,21,26,18,13)(8,22,23,27,25,14,24,30)(19,33,34,28,35,31,32,36);(6,12)(7,18)(8,23)(9,10)(15,16)(17,26)(20,21)(24,25)(27,30)(28,31)(32,34)(33,36);(5,7)(6,13,16,29)(8,19)(9,20)(10,18)(11,21)(12,26,15,17)(14,28,22,36)(23,34)(24,32)(25,35)(27,33,30,31);(2,4)(5,8,9,23,11,25,10,24)(6,14,15,30,16,22,12,27)(7,19,20,34,21,35,18,32)(13,28,17,31,29,36,26,33)

Polycyclic presentation:

2^9|a^2=e,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, 331535);