#114793

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

824 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)(19,28)(20,33,37,25)(21,38)(22,26)(23,27)(24,30)(29,45,47,41)(31,43)(32,40)(34,39,46,48)(35,42,36,44);(1,3)(2,6)(4,12)(5,10)(7,9)(8,16)(11,13)(14,15)(17,19,23,31)(18,24,27,40)(20,34,37,46)(21,32,22,30)(25,41,33,45)(26,43,38,28)(29,35,47,36)(39,44,48,42);(6,9)(14,16)(17,20,23,37)(18,25,27,33)(19,29,31,47)(21,36,22,35)(24,39,40,48)(26,44,38,42)(28,45,43,41)(30,34,32,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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,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, 21989);