#497980

order	512
nilpotency class	3
isoclinism	isoclinic class of #360226
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2352
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22750
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=deh,e^b=ei,g^b=gi,d^c=di,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, 405176);