#288618

order	512
nilpotency class	3
isoclinism	isoclinic class of #288611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 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)(3,10)(4,14)(5,8)(6,9)(7,13)(11,15)(12,16)(17,18,21,24,23,28,22,27)(19,29,30,25,33,40,32,41)(20,34,35,26,37,38,36,42)(31,39,44,47,46,48,45,43);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(18,24,28,27)(19,30,33,32)(26,38,42,34)(31,44,46,45);(2,8)(3,11)(4,6)(7,15)(10,16)(12,13)(17,19)(18,25)(20,31)(21,32)(22,30)(23,33)(24,29)(26,39)(27,40)(28,41)(34,47)(35,45)(36,44)(37,46)(38,43)(42,48);(1,4,5,6)(2,9,8,14)(3,12,13,11)(7,16,10,15)(17,20,22,36,23,37,21,35)(18,26,24,38,28,42,27,34)(19,31,30,44,33,46,32,45)(25,39,29,43,41,48,40,47)

Polycyclic presentation:

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

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, 195814);