#273725

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62087	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,21,27,23,24,22,28)(19,29,33,38,32,40,30,25)(20,34,35,42,36,26,37,41)(31,43,44,39,45,48,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(18,24)(19,30)(21,22)(25,29)(26,34)(31,44)(32,33)(35,37)(38,40)(39,43)(45,46)(47,48);(2,7)(4,13)(6,14)(10,12)(17,19,23,32)(18,25,24,38)(20,31,36,45)(21,30,22,33)(26,39,34,47)(27,40,28,29)(35,46,37,44)(41,43,42,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,34)(25,39)(27,42)(28,41)(29,43)(30,44)(32,45)(33,46)(38,47)(40,48)

Polycyclic presentation:

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

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, 180921);