#114705

order	512
nilpotency class	3
isoclinism	isoclinic class of #114704
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	#60805	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21901);