#278638

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
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

551 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)(3,8)(4,14)(5,12)(6,11)(7,10)(9,15)(13,16)(17,18,23,26,22,28,21,27)(19,29,32,20,30,31,25,24);(1,3,6,12,7,4,15,13)(2,5,9,8,10,16,11,14)(17,19,21,25,22,30,23,32)(18,24,26,29,28,20,27,31);(1,4)(2,8)(3,7)(5,11)(6,13)(9,16)(10,14)(12,15)(17,20)(18,25)(19,27)(21,31)(22,24)(23,29)(26,30)(28,32);(1,5,15,14,7,16,6,8)(2,3,11,13,10,4,9,12)(17,19,23,32,22,30,21,25)(18,24,27,31,28,20,26,29)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=eh,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,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185834);