#273190

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

799 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,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, 180386);