#294526

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

987 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)(3,7)(4,5)(6,8)(9,10,14,11)(12,13,15,16)(17,18,21,25)(19,27,28,24)(20,29,30,32)(22,31,23,26);(2,5)(6,7)(9,11,14,10)(12,16,15,13)(17,19,21,28)(18,24,25,27)(20,23,30,22)(26,29,31,32);(9,12,13,11,14,15,16,10)(17,20,22,28,21,30,23,19)(18,24,26,29,25,27,31,32);(1,3,4,8)(2,6,5,7)(18,25)(24,27)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201722);