#113159

order	512
nilpotency class	3
isoclinism	isoclinic class of #10238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20355);