#497762

order	512
nilpotency class	3
isoclinism	isoclinic class of #360368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2344
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22603
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

493 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:

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

Polycyclic presentation:

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

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, 404958);