#485159

order	512
nilpotency class	4
isoclinism	isoclinic class of #485065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#18162	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

990 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)(4,7)(5,9)(6,11)(10,15)(12,24)(13,19)(14,23)(16,31)(17,20)(18,25)(21,27)(22,28)(26,29)(30,32);(2,7)(4,15)(8,19)(9,10)(11,20)(13,28)(16,30)(17,29)(21,32)(22,23)(24,31)(25,26);(1,3,5,14)(2,8,9,23)(4,16,15,30)(6,12,18,27)(7,19,10,22)(11,24,25,21)(13,29,28,17)(20,31,26,32);(2,9)(3,12)(7,10)(8,21)(11,25)(13,16)(14,27)(19,32)(20,26)(22,31)(23,24)(28,30)

Polycyclic presentation:

2^9|c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392355);