#115814

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6)(7,13,16,10)(8,17)(9,11)(12,19)(14,18,20,15)(21,22,26,30,27,32,25,31)(23,28,35,43,36,44,34,48)(24,37,39,29,40,46,41,47)(33,45,49,42,50,52,38,51);(1,2,3,4)(5,6)(7,14)(8,17)(9,11)(10,15)(12,19)(13,18)(16,20)(21,23)(22,28)(24,38)(25,34)(26,35)(27,36)(29,45)(30,43)(31,48)(32,44)(33,39)(37,51)(40,49)(41,50)(42,46)(47,52);(5,7,9,16)(6,10,11,13)(8,18,19,15)(12,20,17,14)(21,24)(22,29)(23,33)(25,41)(26,39)(27,40)(28,42)(30,46)(31,37)(32,47)(34,38)(35,49)(36,50)(43,52)(44,51)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ehi,f^c=fh,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, 23010);