#469359

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

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

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, 376555);