#347183

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62877	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1955 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254379);