#240113

order	512
nilpotency class	3
isoclinism	isoclinic class of #240110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

810 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)(3,6,10,14)(4,7,12,8)(11,13,16,15)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,37)(28,42)(30,41)(32,46)(34,40)(36,39)(38,43)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19)(18,23)(20,32)(21,36)(22,30)(24,38)(25,41)(26,39)(27,35)(28,33)(29,37)(31,44)(34,43)(40,46)(42,45)(47,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,21,34,22,33,37,48)(18,24,25,40,26,31,35,45)(19,28,29,43,30,32,36,47)(23,38,27,42,39,44,41,46);(3,10)(6,14)(11,16)(13,15)(20,33)(24,31)(28,32)(34,48)(38,44)(40,45)(42,46)(43,47)

Polycyclic presentation:

2^9|a^2=i,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,f^c=fhi,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, 147309);