#202055

order	512
nilpotency class	3
isoclinism	isoclinic class of #201978
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,22,26,23,27,21,37)(19,28,32,39,30,43,31,24)(20,33,36,25,34,41,35,40)(29,38,45,42,46,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,40,48)(26,28,37,43)(33,47,41,42)(34,46)(35,44)(36,45);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,23,34)(18,25,27,40)(19,29,30,46)(21,35,22,36)(24,38,39,48)(26,41,37,33)(28,42,43,47)(31,44,32,45);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(19,30)(20,34)(24,39)(25,40)(28,43)(31,32)(33,41)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109251);