#203981

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111177);