#278619

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	#19496	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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,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, 185815);