#363250

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
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

2464 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)(11,20)(12,24,26,33)(13,27)(14,18)(15,19)(16,22)(17,25,34,29)(21,36,39,30)(23,31)(28,32)(35,38,40,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,31)(19,32)(20,27)(24,35)(25,38)(26,39)(29,37)(33,40)(34,36);(2,5)(6,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,27)(28,38)(31,40)(32,36);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,31)(19,32)(20,27)(24,35)(25,38)(26,39)(29,37)(33,40)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270446);