#273544

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,16)(14,15)(17,18)(19,27,31,24)(20,22)(21,32,35,25)(23,26)(28,30,40,37)(29,38)(33,36)(34,45,47,39)(41,43)(42,44)(46,48);(2,6)(4,13)(7,10)(8,9)(11,14)(15,16)(17,19)(18,24)(20,27)(21,29)(22,31)(23,37)(25,38)(26,40)(28,33)(30,36)(32,41)(34,46)(35,43)(39,48)(42,45)(44,47);(17,20,22,18)(19,28,31,40)(21,32,35,25)(23,26,36,33)(24,37,27,30)(29,42,43,48)(34,39,47,45)(38,46,41,44);(1,3,5,12)(2,7,9,15)(4,14,13,11)(6,10,8,16)(17,21,22,35)(18,25,20,32)(19,29,31,43)(23,34,36,47)(24,38,27,41)(26,39,33,45)(28,42,40,48)(30,44,37,46)

Polycyclic presentation:

2^9|a^2=fhi,c^2=hi,d^2=h,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, 180740);