#479848

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17277
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387044);