#362776

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2292 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)(12,24,27,35)(13,28)(14,18)(15,19)(16,22)(17,26,36,30)(21,34,25,31)(23,32)(29,33)(37,39,40,38);(1,2,4,6)(3,7,8,5)(9,11,15,23)(10,16,19,33)(12,25,27,21)(13,29,14,22)(17,34,36,31)(18,32,28,20)(24,40,35,37)(26,38,30,39);(1,3,4,8)(2,5,6,7)(9,12,15,27)(10,17,19,36)(11,21,23,25)(13,26,14,30)(16,31,33,34)(18,35,28,24)(20,37,32,40)(22,38,29,39);(9,11)(10,16)(12,21)(13,29)(14,22)(15,23)(17,31)(18,32)(19,33)(20,28)(24,37)(25,27)(26,39)(30,38)(34,36)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269972);