#112400

order	512
nilpotency class	3
isoclinism	isoclinic class of #112391
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

766 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)(3,9,11,7)(4,5)(6,13)(8,14)(10,15,16,12)(17,18,22,28,23,27,21,26)(19,29,32,24,33,40,31,39)(20,34,35,42,36,25,37,41)(30,44,45,38,46,43,47,48);(1,3)(2,7)(4,9)(5,11)(6,15)(8,16)(10,13)(12,14)(17,19,23,33)(18,24,27,39)(20,30,36,46)(21,32,22,31)(25,38,34,48)(26,40,28,29)(35,47,37,45)(41,44,42,43);(1,4,5,2)(3,10,11,16)(6,8,14,13)(7,15,9,12)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44)

Polycyclic presentation:

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

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, 19596);