#6468920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,31,21,27,20,26)(19,24,28,23,30,25,29,32);(9,11)(10,13)(12,15)(14,16)(17,19,21,30)(18,23,27,32)(20,28,22,29)(24,26,25,31);(2,5)(6,8)(10,14)(11,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,31)(25,27)(26,32);(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,26)(24,27)(31,32);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,26,27,31)(19,28,30,29)(23,25,32,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6376116);