#469527

order	512
nilpotency class	3
isoclinism	isoclinic class of #454057
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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1223 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376723);