#272655

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	#62132	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

(1,2,6,8,4,9,5,10)(3,11,14,15,13,7,12,16)(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)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(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);(2,8)(3,12)(5,6)(9,10)(13,14)(15,16)(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);(2,9)(7,11)(8,10)(15,16)(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,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, 179851);