#385057

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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)(3,7)(4,6)(5,8)(9,10,14,21,15,20,33,19)(11,22,26,36,25,29,24,16)(12,27,30,37,31,17,32,23)(13,28,34,39,35,38,40,18);(1,3,4,8)(2,5,6,7)(9,11,15,25)(10,16,20,36)(12,28,31,38)(13,17,35,27)(14,26,33,24)(18,30,39,32)(19,29,21,22)(23,40,37,34);(2,6)(5,7)(9,12)(10,17)(11,18)(13,29)(14,30)(15,31)(16,34)(19,37)(20,27)(21,23)(22,35)(24,38)(25,39)(26,28)(32,33)(36,40);(3,8)(5,7)(9,13,33,40,15,35,14,34)(10,18,21,28,20,39,19,38)(11,23,26,27,25,37,24,17)(12,29,32,36,31,22,30,16)

Polycyclic presentation:

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

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, 292253);