#184491

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 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)(3,6)(4,5)(7,8)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(2,5)(3,7)(9,10,11,13,12,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,3,4,7)(2,6,5,8)(18,23)(19,25)(24,30)(26,31);(2,5)(6,8)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=de,c^b=ceh,d^b=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, 91687);