#497279

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₂²⋊C₈
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404475);