#388758

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 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)(4,6)(7,8)(9,10,13,19,15,20,14,18)(11,21,24,16,25,23,29,32)(12,17,27,35,28,36,30,34)(22,33,38,31,39,37,26,40);(1,3,4,7)(2,5,6,8)(9,11,13,24,15,25,14,29)(10,16,18,21,20,32,19,23)(12,26,27,22,28,38,30,39)(17,33,34,40,36,37,35,31);(2,6)(5,8)(9,12,13,27,15,28,14,30)(10,17,19,35,20,36,18,34)(11,22,24,38,25,39,29,26)(16,31,23,37,32,40,21,33);(2,6)(5,8)(9,10)(11,23)(12,17)(13,19)(14,18)(15,20)(16,29)(21,25)(22,37)(24,32)(26,31)(27,35)(28,36)(30,34)(33,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,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,g^b=gi,g^c=gi,e^d=ei

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, 295954);