#269160

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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)(17,18,21,27,23,24,22,28)(19,29,33,38,32,40,30,25)(20,34,36,42,35,26,37,41)(31,43,44,39,46,48,45,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,16)(13,15)(18,24)(19,30)(20,35)(21,22)(25,29)(31,45)(32,33)(38,40)(39,48)(41,42)(43,47)(44,46);(1,4,5,7)(2,3,8,10)(6,14,13,12)(9,15,16,11)(17,19)(18,25)(20,31)(21,30)(22,33)(23,32)(24,38)(26,39)(27,40)(28,29)(34,47)(35,46)(36,45)(37,44)(41,43)(42,48);(2,8)(3,10)(9,16)(11,15)(17,20,23,35)(18,26,24,34)(19,31,32,46)(21,37,22,36)(25,39,38,47)(27,42,28,41)(29,43,40,48)(30,44,33,45)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,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, 176356);