#113676

order	512
nilpotency class	3
isoclinism	isoclinic class of #113669
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

872 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,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, 20872);