#112802

order	512
nilpotency class	3
isoclinism	isoclinic class of #112799
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

795 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)(3,10,12,7)(4,8)(5,9)(6,11,16,13)(14,15)(17,18)(19,28,32,24)(20,33,37,44)(21,27)(22,38)(23,26)(25,35,42,36)(29,46)(30,43,31,40)(34,45)(39,47)(41,48);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19,23,32)(18,24,26,28)(20,34,37,47)(21,31,22,30)(25,41,42,46)(27,43,38,40)(29,35,48,36)(33,39,44,45);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20,23,37)(18,25,26,42)(19,29,32,48)(21,36,22,35)(24,39,28,45)(27,44,38,33)(30,34,31,47)(40,46,43,41)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19998);