#383701

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 290897);