#273813

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
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	#62090	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 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)(3,9,11,15)(4,7,13,14)(5,10)(6,8)(12,16)(17,18,24,27,23,21,22,28)(19,29,32,25,34,42,33,31)(20,35,38,26,40,44,39,37)(30,41,36,43,47,45,46,48);(1,3,6,11)(2,7,8,14)(4,12,13,5)(9,10,15,16)(17,19)(18,25)(20,36)(21,31)(22,33)(23,34)(24,32)(26,43)(27,42)(28,29)(30,39)(35,41)(37,48)(38,47)(40,46)(44,45);(2,8)(3,11)(5,12)(9,15)(17,20,23,40)(18,26,21,37)(19,30,34,47)(22,38,24,39)(25,41,31,45)(27,35,28,44)(29,43,42,48)(32,46,33,36);(1,4)(2,9)(3,12)(5,11)(6,13)(7,10)(8,15)(14,16)(17,21)(18,23)(19,31)(20,37)(22,27)(24,28)(25,34)(26,40)(29,32)(30,45)(33,42)(35,38)(36,48)(39,44)(41,47)(43,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181009);