#283743

order	512
nilpotency class	3
isoclinism	isoclinic class of #283707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,23,27,24,29,22,28)(19,30,32,25,20,34,33,41)(21,36,37,26,39,43,38,44)(31,40,47,48,35,42,46,45);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19,23,32,24,20,22,33)(18,25,27,34,29,41,28,30)(21,35,37,46,39,31,38,47)(26,42,43,45,44,40,36,48);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,24,19)(18,25,29,41)(21,35,39,31)(22,33,23,32)(26,40,44,42)(27,30,28,34)(36,48,43,45)(37,47,38,46);(17,21)(18,26)(19,31)(20,35)(22,38)(23,37)(24,39)(25,40)(27,43)(28,36)(29,44)(30,45)(32,47)(33,46)(34,48)(41,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190939);