#112644

order	512
nilpotency class	3
isoclinism	isoclinic class of #112637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

772 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)(3,10,12,7)(4,8)(5,9)(6,11,16,13)(14,15)(17,18,22,28,23,26,21,27)(19,29,32,39,33,24,31,40)(20,25,35,43,36,42,37,44)(30,45,34,48,46,38,47,41);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19,23,33)(18,24,26,29)(20,34,36,47)(21,32,22,31)(25,41,42,48)(27,40,28,39)(30,37,46,35)(38,44,45,43);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20,23,36)(18,25,26,42)(19,30,33,46)(21,37,22,35)(24,38,29,45)(27,44,28,43)(31,47,32,34)(39,48,40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19840);