#202532

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109728);