#347889

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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:

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

Polycyclic presentation:

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

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, 255085);