#6465307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6372503);