#107894

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 15090);