#223373

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

802 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)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23)(25,26,28,32,29,31,35,38)(27,30,34,37,33,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27,29,33)(26,30,31,36)(28,34,35,39)(32,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,18,20)(11,19,13,21)(15,23,17,24)(27,33)(30,36)(34,39)(37,40);(2,5)(6,8)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(26,31)(30,36)(32,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=eh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130569);