#6461205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368401);