#482774

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	#40590	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

1127 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)(13,27)(14,24)(15,23,28,21)(16,26)(25,30,32,29);(2,7)(3,13)(4,18)(8,23)(9,11)(10,22)(12,17)(14,16)(15,29)(19,31)(21,27)(24,25)(26,32)(28,30);(1,3,6,13)(2,8,12,27)(4,15,19,28)(5,16,20,14)(7,21,17,23)(9,25,22,32)(10,24,11,26)(18,30,31,29);(3,14)(8,24)(13,16)(15,29)(21,32)(23,25)(26,27)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389970);