#274919

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 182115);