#269185

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	#60920	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^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, 176381);