#347471

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63061	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1950 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^d=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, 254667);