#112917

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

890 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,g^2=i,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, 20113);