#238848

order	512
nilpotency class	3
isoclinism	isoclinic class of #238843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 146044);