#247825

order	512
nilpotency class	3
isoclinism	isoclinic class of #247783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1075 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155021);