#273269

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180465);