#105645

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ci,f^b=fh,e^c=eh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12841);