#278621

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19484	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 185817);