#429999

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

1528 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)(6,11,14,22)(12,26,15,25)(13,19,28,33)(27,36,29,30);(5,6,9,22,10,14,24,11)(7,15,17,26,18,12,23,25)(8,19,20,13,21,33,35,28)(16,30,31,29,32,36,34,27);(5,7,10,18)(6,12,14,15)(8,16,21,32)(9,23,24,17)(11,25,22,26)(13,27,28,29)(19,30,33,36)(20,34,35,31);(2,4)(5,8)(6,13,14,28)(7,16)(9,20)(10,21)(11,19,22,33)(12,27,15,29)(17,31)(18,32)(23,34)(24,35)(25,30,26,36)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 337195);