#387934

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=egi,d^2=eg,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295130);