#294782

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

900 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)(9,10,14,16)(11,15,13,12)(17,18,22,26)(19,28,29,24)(20,30,31,32)(21,27,23,25);(2,5)(6,7)(9,11,14,13)(10,15,16,12)(17,19,22,29)(18,24,26,28)(20,23,31,21)(25,30,27,32);(9,12,13,10,14,15,11,16)(17,20,23,29,22,31,21,19)(18,24,27,30,26,28,25,32);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,25,26,27)(19,20,29,31)(24,30,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201978);