#423423

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1405 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)(7,11)(8,12,19,28)(9,21)(10,13)(14,23)(15,24,31,36)(16,32)(17,25)(18,35,33,27)(20,30,22,29)(26,34);(5,7,10,17)(6,11,13,25)(8,18)(9,16,23,34)(12,27)(14,26,21,32)(15,22)(19,33)(20,31)(24,30)(28,35)(29,36);(5,8)(6,12,13,28)(7,15)(9,22)(10,19)(11,24,25,36)(14,30,21,29)(16,33)(17,31)(18,34)(20,23)(26,35,32,27);(2,4)(9,23)(14,21)(16,34)(18,33)(20,22)(26,32)(27,35)(29,30)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ef,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^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, 330619);