#474323

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16953
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

653 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)(3,5,15,8)(4,9)(6,11)(7,12)(10,19,21,23)(13,16,30,24)(14,20)(17,27)(18,26)(22,29)(25,31,28,32);(1,3,7,21,6,15,20,10)(2,8,12,19,11,5,14,23)(4,13,18,28,17,30,22,25)(9,16,26,32,27,24,29,31);(1,4,6,17)(2,9)(3,13,15,30)(5,16)(7,22,20,18)(8,24)(10,28,21,25)(11,27)(12,29)(14,26)(19,32)(23,31);(1,5,6,8)(2,10,11,21)(3,14,15,12)(4,16)(7,23,20,19)(9,25)(13,26)(17,24)(18,32)(22,31)(27,28)(29,30)

Polycyclic presentation:

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

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, 381519);