#273682

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,14)(13,16)(17,18)(19,27,31,24)(20,22)(21,32,36,25)(23,26)(28,30,42,38)(29,39)(33,37)(34,40)(35,45,48,41)(43,44)(46,47);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,12,10)(17,19,22,31)(18,24,20,27)(21,34,36,29)(23,38,37,30)(25,40,32,39)(26,42,33,28)(35,44,48,47)(41,43,45,46);(1,4,7,3)(2,9,11,12)(5,14,6,13)(8,15,16,10)(17,20,22,18)(19,28,31,42)(21,32,36,25)(23,26,37,33)(24,38,27,30)(29,43,34,46)(35,41,48,45)(39,47,40,44);(1,5)(3,13)(4,14)(6,7)(8,16)(10,15)(17,21,22,36)(18,25,20,32)(19,29,31,34)(23,35,37,48)(24,39,27,40)(26,41,33,45)(28,43,42,46)(30,44,38,47)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=h,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180878);