#363948

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2590 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)(11,22)(12,25,28,35)(13,17)(14,30)(15,20)(16,21)(18,27,36,31)(19,24)(23,38,40,32)(26,37,39,34)(29,33);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,23)(13,15)(14,29)(16,24)(18,32)(19,20)(21,33)(22,30)(25,37)(26,31)(27,39)(28,40)(34,35)(36,38);(2,5)(6,8)(9,12)(10,18)(11,23)(13,26)(14,27)(15,31)(16,28)(17,32)(19,34)(20,35)(21,36)(22,37)(24,40)(25,30)(29,39)(33,38);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,14)(12,26)(15,24)(16,29)(17,30)(18,34)(20,33)(21,22)(23,27)(25,32)(28,39)(31,40)(35,38)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^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, 271144);