#112009

order	512
nilpotency class	3
isoclinism	isoclinic class of #112003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#37550	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 19205);