#358916

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 266112);