#263782

order	512
nilpotency class	3
isoclinism	isoclinic class of #263779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18276	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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:

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

Polycyclic presentation:

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

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, 170978);