#184633

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=chi,d^a=de,c^b=ce,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, 91829);