#269165

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	#18304	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,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, 176361);