#331258

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

671 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,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, 238454);