#114781

order	512
nilpotency class	3
isoclinism	isoclinic class of #114778
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×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

832 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:

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

Polycyclic presentation:

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

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, 21977);