#316925

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224121);