#347478

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
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	#63063	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2093 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:

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

Polycyclic presentation:

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

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, 254674);