#108150

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

722 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)(5,6,11,14,9,8,10,13)(7,15,17,19,18,12,16,20)(21,22,26,30,27,31,25,41)(23,32,36,43,35,47,34,28)(24,37,39,45,40,29,38,44)(33,46,50,42,49,52,48,51);(2,4)(5,7,10,17,9,18,11,16)(6,12,14,19,8,15,13,20)(21,23)(22,28,31,43)(24,33)(25,34)(26,36)(27,35)(29,42,37,51)(30,32,41,47)(38,48)(39,50)(40,49)(44,52,45,46);(5,8,9,6)(7,15,18,12)(10,13,11,14)(16,19,17,20)(21,24,27,40)(22,29,31,37)(23,33,35,49)(25,39,26,38)(28,42,43,51)(30,45,41,44)(32,46,47,52)(34,50,36,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 15346);