#6460323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,26,30,32)(20,23,31,25)(22,28,27,29);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,25)(21,24)(22,30)(23,31)(26,28)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,20,29,21,30,31,28)(18,22,25,26,24,27,23,32);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,20,21,31)(18,23,24,25)(19,28,30,29)(22,26,27,32);(17,20,21,31)(18,23,24,25)(19,29,30,28)(22,32,27,26)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6367519);