#272621

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	#62145	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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^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, 179817);