#283713

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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 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:

(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);(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(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);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(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);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(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=f,b^2=f,c^2=h,d^2=i,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, 190909);