#273445

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
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	#61415	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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)(3,8)(4,12)(5,7)(6,10)(9,11)(13,15)(14,16)(17,18)(19,27,29,23)(20,24)(21,26)(22,25)(28,37,46,45)(30,43,34,38)(31,39,47,48)(32,41)(33,40)(35,44)(36,42);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,19)(18,23)(20,31)(21,34)(22,29)(24,39)(25,27)(26,43)(28,33)(30,36)(32,47)(35,46)(37,40)(38,42)(41,48)(44,45);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,41)(26,44)(27,45)(29,46)(30,47)(31,34)(33,36)(38,48)(39,43)(40,42);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,37)(25,27)(26,38)(31,33)(32,46)(34,36)(35,47)(39,40)(41,45)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180641);