#107458

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14654);