#114756

order	512
nilpotency class	3
isoclinism	isoclinic class of #114746
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

857 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,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, 21952);