#347895

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

809 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)(19,28)(21,24)(26,36)(27,37)(29,44)(30,31)(32,33)(35,39)(38,47)(41,42)(43,45);(17,18,24,37,22,27,21,23)(19,25,28,39,31,40,30,35)(20,26,33,41,34,42,32,36)(29,38,43,48,45,47,44,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,25)(20,29)(21,30)(22,31)(23,35)(24,28)(26,38)(27,40)(32,44)(33,43)(34,45)(36,46)(37,39)(41,48)(42,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,34)(18,26,27,42)(19,29,31,45)(21,32,24,33)(23,36,37,41)(25,38,40,47)(28,43,30,44)(35,46,39,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,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, 255091);