#285434

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

600 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)(4,11)(5,7)(8,12)(9,15)(10,14)(13,16)(17,18,22,27,23,28,21,26)(19,24,31,40,32,41,30,39)(20,33,35,43,36,44,37,25)(29,45,46,42,34,48,47,38);(2,7)(3,9)(5,15)(8,11)(10,16)(12,14)(17,19,22,31,23,32,21,30)(18,24,27,40,28,41,26,39)(20,34,35,47,36,29,37,46)(25,42,33,48,43,38,44,45);(1,3,6,2)(4,10,13,8)(5,7,15,9)(11,12,16,14)(17,20,23,36)(18,25,28,43)(19,29,32,34)(21,35,22,37)(24,38,41,42)(26,33,27,44)(30,46,31,47)(39,45,40,48);(1,4,6,13)(2,8,3,10)(5,14,15,12)(7,11,9,16)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,35,36,37)(24,39,41,40)(25,33,43,44)(29,46,34,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,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, 192630);