#494136

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

574 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)(9,10,15,20,13,21,29,38,14,22,31,18,30,19,12,27)(11,17,26,36,24,16,23,33,25,34,32,39,40,35,28,37);(2,5)(6,8)(9,11,14,25)(10,16)(12,28,29,23)(13,24,30,40)(15,26,31,32)(17,19)(18,37)(20,33)(21,34)(22,35)(27,36)(38,39);(9,11)(10,17)(12,23)(13,24)(14,25)(15,32)(16,21)(18,36)(19,35)(20,39)(22,34)(26,31)(27,33)(28,29)(30,40)(37,38);(1,3,4,7)(2,6,5,8)(9,12,13,15,14,29,30,31)(10,18,19,38,22,20,21,27)(11,23,24,32,25,28,40,26)(16,33,17,36,35,37,34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=ci,d^b=di,g^c=gi,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401332);