#247805

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	#42788	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1079 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,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, 155001);