#471074

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16744
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378270);