#391483

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,26,22,27,21,28)(19,20,30,31,29,24,25,32);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19)(18,24)(20,27)(21,25)(22,29)(23,30)(26,32)(28,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,24)(18,25,27,30)(19,28,29,26)(21,32,23,31);(2,5)(3,7)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,28)(19,25,29,30)(20,31,24,32)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

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, 298679);