#493178

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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)(11,23,26,33)(12,29)(13,20)(14,32)(15,22)(16,18)(17,24,35,28)(19,30)(21,31)(25,36,39,37)(27,40,38,34);(2,5)(6,8)(9,11,14,27,15,28,30,39,16,26,31,38,13,24,12,25)(10,17,21,36,22,33,29,40,18,35,32,37,20,23,19,34);(10,18)(11,24,26,28)(12,30)(14,31)(17,33,35,23)(20,22)(25,27,39,38)(34,37,40,36);(1,3,4,7)(2,6,5,8)(9,12,13,31,16,30,15,14)(10,19,20,32,18,29,22,21)(11,25,24,38,26,39,28,27)(17,34,23,37,35,40,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400374);