#330142

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10,15,23,13,18,14,19)(11,20,22,24,12,16,21,17)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,2,4,5)(3,6,7,8)(9,11,13,12)(10,16,18,20)(14,21,15,22)(17,19,24,23)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,17,18,24)(14,21,15,22)(16,19,20,23)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(3,7)(6,8)(10,18)(11,12)(19,23)(21,22)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ei,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237338);