#501028

order	512
nilpotency class	4
isoclinism	isoclinic class of #501028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408224);