#112371

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

884 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:

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

Polycyclic presentation:

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

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, 19567);