#272564

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

800 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)(20,34)(21,38)(22,26)(23,27)(24,31)(25,36)(29,33)(30,40)(32,41)(35,42)(37,43)(39,44)(45,47)(46,48);(1,3,8,13)(2,9,12,14)(4,6,5,7)(10,11,16,15)(17,19,23,33)(18,24,27,41)(20,35,37,30)(21,32,22,31)(25,42,44,40)(26,29,38,28)(34,46,43,45)(36,47,39,48);(1,4,8,5)(2,10,12,16)(3,6,13,7)(9,11,14,15)(17,18)(19,29)(20,25)(21,26)(22,38)(23,27)(24,31)(28,33)(30,46)(32,41)(34,36)(35,45)(37,44)(39,43)(40,48)(42,47);(1,5,8,4)(2,10,12,16)(3,6,13,7)(9,15,14,11)(17,20,23,37)(18,25,27,44)(19,30,33,35)(21,39,22,36)(24,40,41,42)(26,43,38,34)(28,45,29,46)(31,48,32,47)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^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, 179760);