#451310

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15994
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358506);