#107258

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

794 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)(7,14,16,11)(8,10)(9,13)(12,17)(15,19,20,18)(21,22,25,30,26,29,40,44)(23,31,34,45,33,27,38,42)(24,35,37,46,36,28,39,43)(32,47,49,52,48,41,50,51);(1,2,3,4)(6,10)(8,17)(11,18)(12,13)(14,19)(15,20)(21,23)(22,27)(24,32)(25,38)(26,33)(28,41)(29,31)(30,45)(34,40)(35,47)(36,48)(37,50)(39,49)(42,44)(43,52)(46,51);(5,7)(6,11)(8,15)(9,16)(10,18)(12,19)(13,14)(17,20)(21,24)(22,28)(23,32)(25,39)(26,36)(27,41)(29,35)(30,46)(31,47)(33,48)(34,50)(37,40)(38,49)(42,52)(43,44)(45,51)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14454);