#272657

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
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	#61998	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh

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, 179853);