#371306

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 278502);