#376203

order	512
nilpotency class	3
isoclinism	isoclinic class of #376201
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14175
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283399);