#269175

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 176371);