#272633

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	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,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, 179829);