#112915

order	512
nilpotency class	3
isoclinism	isoclinic class of #112912
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

895 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18,22,27,23,26,21,36)(19,28,31,43,32,24,30,39)(20,33,34,25,35,42,37,41)(29,44,45,40,46,48,47,38);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19,23,32)(18,24,26,28)(20,29,35,46)(21,31,22,30)(25,40,41,38)(27,43,36,39)(33,48,42,44)(34,47,37,45);(6,14)(9,15)(17,20)(18,25)(19,29)(21,37)(22,34)(23,35)(24,38)(26,41)(27,42)(28,40)(30,47)(31,45)(32,46)(33,36)(39,44)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20111);