#118157

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

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

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, 25353);