#448602

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355798);