#289424

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

535 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,d^2=ehi,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,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196620);