#482723

order	512
nilpotency class	4
isoclinism	isoclinic class of #482722
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#35922	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#551 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1068 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)(3,8)(4,17,19,7)(5,11)(6,12)(9,18,22,31)(10,20)(13,23)(14,28,30,21)(15,25)(16,26)(24,29,32,27);(2,7)(4,18)(8,21)(9,11)(10,22)(12,17)(14,29)(19,31)(23,32)(24,25)(26,28)(27,30);(1,3,6,16)(2,8,12,26)(4,14,19,30)(5,15,20,13)(7,21,17,28)(9,24,22,32)(10,23,11,25)(18,29,31,27);(3,13)(8,23)(14,27)(15,16)(21,32)(24,28)(25,26)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389919);