#362337

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269533);