#273369

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#525	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

878 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)(3,9,11,7)(4,6)(5,12,13,8)(10,14)(15,16)(18,24,27,28)(20,34,35,36)(25,38,40,29)(30,44,45,46);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,32)(30,43)(31,40)(33,38)(34,37)(35,42)(36,41)(39,44)(45,47)(46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19,23,33)(18,25,27,40)(20,30,35,45)(21,32,22,31)(24,29,28,38)(26,39,42,48)(34,46,36,44)(37,43,41,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,20,23,35)(18,26,27,42)(19,30,33,45)(21,36,22,34)(24,37,28,41)(25,39,40,48)(29,43,38,47)(31,46,32,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,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, 180565);