#482191

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1167 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)(3,13,14,8)(4,17,19,7)(5,11)(6,12)(9,18,22,31)(10,20)(15,21)(16,24,29,25)(23,30)(26,28)(27,32);(2,7)(4,18)(8,21)(9,11)(10,22)(12,17)(13,26)(15,30)(19,31)(23,25)(24,27)(28,32);(1,3,5,16)(2,8,11,25)(4,15,18,30)(6,14,20,29)(7,21,9,23)(10,24,12,13)(17,26,22,27)(19,28,31,32);(3,14)(8,13)(15,28)(16,29)(21,26)(23,27)(24,25)(30,32)

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389387);