#184707

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=de,c^b=cei,d^b=dh,d^c=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91903);