#424340

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

1612 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)(5,6,9,14,10,15,22,11)(7,16,19,12,20,26,17,27)(8,13,21,28,23,29,34,24)(18,30,32,25,33,35,31,36);(6,11)(7,17)(8,21)(9,22)(14,15)(16,26)(19,20)(23,34)(24,28)(25,35)(30,36)(31,32);(5,7,10,20)(6,12)(8,18,23,33)(9,19,22,17)(11,16)(13,25)(14,26)(15,27)(21,32,34,31)(24,30)(28,35)(29,36);(2,4)(5,8,22,34,10,23,9,21)(6,13,11,24,15,29,14,28)(7,18,17,31,20,33,19,32)(12,25,16,30,27,36,26,35)

Polycyclic presentation:

2^9|a^2=e,c^2=ef,d^2=ei,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, 331536);