#356066

order	512
nilpotency class	3
isoclinism	isoclinic class of #356055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊C₄²
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2319 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263262);