#278556

order	512
nilpotency class	3
isoclinism	isoclinic class of #278523
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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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)(3,9)(4,15)(5,14)(6,12)(7,11)(8,13)(10,16)(17,18,24,27,23,20,22,28)(19,26,21,31,30,32,29,25);(1,3,6,13,7,4,16,14)(2,8,10,9,11,5,12,15)(17,19,22,29,23,30,24,21)(18,25,27,26,20,31,28,32);(1,4)(2,9)(3,7)(5,10)(6,14)(8,12)(11,15)(13,16)(17,20,23,18)(19,25,30,31)(21,32,29,26)(22,28,24,27);(1,5,6,15,7,8,16,9)(2,4,10,14,11,3,12,13)(17,21,24,30,23,29,22,19)(18,26,28,25,20,32,27,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,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,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185752);