#243081

order	512
nilpotency class	3
isoclinism	isoclinic class of #243076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

599 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150277);