#432174

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1383 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)(6,12,15,23)(13,26,28,16)(14,20,29,30)(27,35,36,31);(5,6,9,23,11,15,10,12)(7,16,18,28,19,26,24,13)(8,20,21,14,22,30,25,29)(17,27,32,31,34,36,33,35);(5,7,11,19)(6,13)(8,17,22,34)(9,24,10,18)(12,16)(14,27)(15,28)(20,31)(21,33,25,32)(23,26)(29,36)(30,35);(2,4)(5,8)(6,14,15,29)(7,17)(9,21)(10,25)(11,22)(12,20,23,30)(13,27,28,36)(16,31,26,35)(18,32)(19,34)(24,33)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=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, 339370);