#363227

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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

2450 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,h^c=hi,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, 270423);