#448145

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

928 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)(10,13)(14,15)(17,18,20,24,21,23,29,31)(19,25,27,30,28,32,26,22);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,15,16,14)(17,19,21,28)(18,22,23,30)(20,26,29,27)(24,32,31,25);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15);(1,4,5,3)(2,7,8,6)(10,13)(14,15)(18,23)(19,26)(20,29)(22,25)(27,28)(30,32)

Polycyclic presentation:

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

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, 355341);