#482717

order	512
nilpotency class	4
isoclinism	isoclinic class of #482716
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	#35683	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

1077 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)(4,16,18,7)(5,11)(6,12)(9,17,21,30)(10,19)(13,22)(14,28,29,20)(15,25)(23,26,31,32)(24,27);(2,7)(4,17)(8,20)(9,11)(10,21)(12,16)(14,26)(18,30)(22,23)(24,31)(25,28)(29,32);(1,3,6,15)(2,8,12,25)(4,14,18,29)(5,13,19,27)(7,20,16,28)(9,23,21,31)(10,24,11,22)(17,26,30,32);(3,13)(8,22)(14,26)(15,27)(20,23)(24,25)(28,31)(29,32)

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=dh,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, 389913);