#448110

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355306);