#383050

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1998 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)(3,4)(5,6,7,8)(9,10,13,19,14,20,15,16)(11,21,22,30,24,17,29,33)(12,25,26,18,27,31,28,34)(23,37,38,35,39,40,36,32);(10,16)(11,22)(12,26)(13,15)(18,34)(19,20)(24,29)(27,28)(30,33)(32,37)(35,40)(36,38);(2,4)(9,11,14,24)(10,17,20,21)(12,23,27,39)(13,22,15,29)(16,30,19,33)(18,32,34,35)(25,40,31,37)(26,38,28,36);(2,4)(9,12)(10,18)(11,23)(13,28)(14,27)(15,26)(16,31)(17,32)(19,25)(20,34)(21,35)(22,36)(24,39)(29,38)(30,37)(33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,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, 290246);