#273224

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
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	#3522	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180420);