#461746

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 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)(3,4)(6,7)(9,10,14,13)(11,16,12,15)(17,18,21,22)(19,26,28,32)(20,24,30,25)(23,27,31,29);(5,6,8,7)(9,11,14,12)(10,15,13,16)(18,22)(23,31)(24,25)(26,32);(5,6)(7,8)(9,12,14,11)(10,15,13,16)(17,19,20,29,21,28,30,27)(18,23,25,26,22,31,24,32);(2,4)(5,7,8,6)(9,13,14,10)(11,16,12,15)(17,20,21,30)(18,24,22,25)(19,27,28,29)(23,26,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368942);