#299248

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60797	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

933 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)(3,5)(4,11)(6,10)(7,13)(8,14)(9,15)(12,16)(17,18,23,26)(19,27,31,39)(20,22,33,37)(21,32,36,48)(24,30,40,29)(25,41,43,35)(28,42,34,38)(44,46,47,45);(1,3,5,2)(4,9,12,7)(6,10,14,8)(11,13,16,15)(17,19,22,29,23,31,37,30)(18,24,20,27,26,40,33,39)(21,34,35,46,36,28,41,45)(25,42,32,47,43,38,48,44);(1,4,6,13,5,12,14,15)(2,7,8,11,3,9,10,16)(17,20,22,26,23,33,37,18)(19,27,30,24,31,39,29,40)(21,32,35,43,36,48,41,25)(28,44,46,38,34,47,45,42);(2,3)(7,9)(8,10)(11,16)(17,21,22,35,23,36,37,41)(18,25,20,32,26,43,33,48)(19,28,29,45,31,34,30,46)(24,38,27,44,40,42,39,47)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206444);