#421530

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

969 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)(6,11)(8,19)(9,22)(12,24)(13,26)(14,23)(15,28)(16,30)(17,33)(21,27)(25,29)(32,36);(5,6,10,14)(7,15)(8,20,21,35)(9,23,22,11)(12,18)(13,27,26,19)(16,31)(17,24)(25,36)(28,33)(29,30)(32,34);(5,7,9,17,10,18,22,33)(6,12,11,24,14,15,23,28)(8,16,19,30,21,32,27,36)(13,25,20,34,26,29,35,31);(2,4)(5,8,9,19,10,21,22,27)(6,13,11,20,14,26,23,35)(7,16,17,30,18,32,33,36)(12,25,24,34,15,29,28,31)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=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, 328726);