#363274

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	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2454 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)(11,21)(12,24,27,17)(13,28)(14,30)(15,20)(16,23)(18,29)(19,32)(22,37,25,40)(26,35,31,36)(33,38,34,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,25)(13,15)(14,29)(17,34)(18,20)(19,28)(21,30)(22,27)(23,32)(24,33)(26,38)(31,39)(35,40)(36,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,25)(14,31)(15,27)(16,33)(18,34)(19,36)(20,24)(21,37)(23,39)(26,32)(28,40)(29,38)(30,35);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,18,20,16)(12,25,27,22)(14,23,32,29)(17,34,24,33)(19,21,30,28)(26,38,31,39)(35,40,36,37)

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,h^c=hi,e^d=ei,f^d=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, 270470);