#184415

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=ci,d^b=dh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91611);