#363496

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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:

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

Polycyclic presentation:

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

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, 270692);