#423744

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi

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, 330940);