#320478

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

734 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)(3,7)(4,5)(6,8)(9,10,13,15,12,14,11,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(2,5)(6,7)(10,14)(15,16)(17,19,21,27,20,26,28,25)(18,22,24,29,23,30,31,32);(9,10)(11,15)(12,14)(13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,3,4,8)(2,6,5,7)(9,11,12,13)(10,15,14,16)(19,26)(22,30)(25,27)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227674);