#422777

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1932 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)(5,6,10,14)(7,15,19,33)(8,12,21,30)(9,22,24,13)(11,25,28,18)(16,34,36,27)(17,32,35,26)(20,31,23,29);(5,7,9,18,10,19,24,25)(6,11,13,15,14,28,22,33)(8,16,20,35,21,36,23,17)(12,26,29,34,30,32,31,27);(5,8)(6,12)(7,16,19,36)(9,23)(10,21)(11,26,28,32)(13,31)(14,30)(15,27,33,34)(17,25,35,18)(20,24)(22,29);(2,4)(5,8)(6,12)(7,17)(9,23)(10,21)(11,27)(13,31)(14,30)(15,32)(16,25)(18,36)(19,35)(20,24)(22,29)(26,33)(28,34)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=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, 329973);