#434837

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

951 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,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, 342033);