#6464724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,26)(21,25)(22,24)(23,27)(29,32)(30,31);(9,10)(11,15)(12,13)(14,16)(17,19,22,25,23,30,20,29)(18,21,26,28,27,32,24,31);(2,5)(6,8)(10,13)(14,15)(17,20,23,22)(18,24,27,26)(19,25,30,29)(21,28,32,31);(9,11,12,16)(10,14,13,15)(17,21,23,32)(18,25,27,29)(19,26,30,24)(20,31,22,28);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,26,27,24)(19,29,30,25)(21,28,32,31)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,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, 6371920);