#359020

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(5,6,8,7)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,23)(17,28)(22,37)(24,32)(25,33)(27,34)(29,35)(30,36)(31,39)(38,40);(2,4)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,32)(19,33)(20,21)(22,29)(26,37)(28,39)(30,38)(31,36)(35,40);(9,12,15,30)(10,17,18,35)(11,22,25,39)(13,29,14,28)(16,31,32,37)(19,36,20,26)(21,34,33,40)(23,27,24,38);(5,7,8,6)(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,37)(34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266216);