#289371

order	512
nilpotency class	3
isoclinism	isoclinic class of #289363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196567);