#323837

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eghi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,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, 231033);