#273497

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	C₂³	D₄²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(3,11)(4,13)(5,15)(6,8)(7,12)(9,14)(10,16)(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);(1,3,6,4)(2,7,8,9)(5,14,16,12)(10,13,15,11)(17,19,22,31)(18,24,20,27)(21,29,35,43)(23,37,36,30)(25,38,32,41)(26,40,33,28)(34,46,47,44)(39,48,45,42);(2,8)(7,9)(10,15)(11,13)(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);(1,4,6,3)(2,9,8,7)(5,12,16,14)(10,11,15,13)(17,21)(18,25)(19,29)(20,32)(22,35)(23,34)(24,38)(26,39)(27,41)(28,42)(30,44)(31,43)(33,45)(36,47)(37,46)(40,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,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, 180693);