#293421

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1039 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,23,26,32)(20,28,21,27)(24,31,25,30);(9,10)(11,15,16,13)(12,14)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(1,3,4,8)(2,6,5,7)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,31,32,30)

Polycyclic presentation:

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

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, 200617);