#6457045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364241);