#119403

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

534 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)(8,12)(9,19)(10,14)(11,16)(13,18)(17,20)(21,22,26,32,27,30,25,31)(23,33,36,43,37,28,35,44)(24,38,39,46,40,47,41,29)(34,48,49,42,50,52,51,45);(1,2)(3,4)(5,7,10,17)(6,11,14,12)(8,18,16,9)(13,20,19,15)(21,23,27,37)(22,28,30,33)(24,34,40,50)(25,36,26,35)(29,45,46,42)(31,44,32,43)(38,52,47,48)(39,51,41,49);(2,4)(5,8,9,7,10,16,18,17)(6,12,13,20,14,11,19,15)(21,24)(22,29,30,46)(23,34)(25,39)(26,41)(27,40)(28,42,33,45)(31,38,32,47)(35,49)(36,51)(37,50)(43,52,44,48)

Polycyclic presentation:

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

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, 26599);