#363251

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

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

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,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, 270447);