#112008

order	512
nilpotency class	3
isoclinism	isoclinic class of #112003
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

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,g^2=i,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, 19204);