#479864

order	512
nilpotency class	3
isoclinism	isoclinic class of #479863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17290
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=ef,d^2=ei,f^2=i,g^2=i,h^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,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387060);