#115528

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

835 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)(3,10,13,7)(4,6)(5,9)(8,14)(11,15,12,16)(17,18,22,26,23,27,21,34)(19,24,29,39,30,40,35,38)(20,25,33,44,32,42,36,43)(28,37,46,41,45,48,31,47);(2,6)(3,11)(4,14)(8,9)(12,13)(15,16)(17,19)(18,24)(20,31)(21,35)(22,29)(23,30)(25,41)(26,39)(27,40)(28,33)(32,46)(34,38)(36,45)(37,43)(42,47)(44,48);(1,3,5,13)(2,7,9,10)(4,12,14,11)(6,15,8,16)(17,20)(18,25)(19,28)(21,36)(22,33)(23,32)(24,37)(26,44)(27,42)(29,46)(30,45)(31,35)(34,43)(38,47)(39,41)(40,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

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, 22724);