#112596

order	512
nilpotency class	3
isoclinism	isoclinic class of #112591
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

818 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19792);