#111998

order	512
nilpotency class	3
isoclinism	isoclinic class of #111989
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

784 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)(3,11,4,7)(5,10)(6,9)(8,13,15,12)(14,16)(17,18,22,28,23,27,21,26)(19,29,32,24,33,39,31,38)(20,25,34,41,35,42,36,43)(30,44,45,37,46,48,47,40);(1,3)(2,7)(4,6)(5,12)(8,16)(9,11)(10,15)(13,14)(17,19,23,33)(18,24,27,38)(20,30,35,46)(21,32,22,31)(25,40,42,37)(26,39,28,29)(34,47,36,45)(41,48,43,44);(1,4,6,3)(2,8,9,15)(5,12,14,13)(7,10,11,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,43)(27,42)(28,41)(29,44)(31,47)(32,45)(33,46)(38,40)(39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19194);