#361001

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
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

1255 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)(11,19,22,15)(12,16)(13,18)(14,17)(20,37,38,29)(21,35,26,30)(23,40,39,31)(24,33)(25,32)(27,36)(28,34);(9,11)(10,15)(12,23)(13,26)(14,22)(16,31)(17,19)(18,35)(20,24)(21,28)(25,39)(27,38)(29,33)(30,34)(32,40)(36,37);(2,5)(6,8)(9,12,14,25)(10,16,17,32)(11,20,22,38)(13,27,28,24)(15,29,19,37)(18,36,34,33)(21,39,26,23)(30,40,35,31);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,24)(25,38)(26,28)(27,39)(31,33)(32,37)(34,35)(36,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 268197);