#423424

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1367 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)(5,6,9,11)(7,12)(8,13,18,25)(10,21,20,14)(15,26)(16,27)(17,24)(19,30,23,29)(22,28)(31,35)(32,34)(33,36);(6,11)(7,15,17,22)(8,18)(12,26,24,28)(14,21)(16,31,32,33)(19,23)(27,35,34,36);(5,7,9,17)(6,12)(8,16,18,32)(10,22,20,15)(11,24)(13,27)(14,26)(19,31,23,33)(21,28)(25,34)(29,36)(30,35);(2,4)(5,8,9,18)(6,13,11,25)(7,16,17,32)(10,23,20,19)(12,27,24,34)(14,30,21,29)(15,31,22,33)(26,35,28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ef,d^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^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, 330620);