#6449657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	#34889	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,29,31,25)(20,26,30,32)(21,27,23,28);(2,5)(6,8)(18,24)(19,30)(20,31)(21,23)(25,32)(26,29);(10,13)(14,16)(17,19,22,31)(18,25,24,29)(20,21,30,23)(26,28,32,27);(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,21)(22,30)(23,31)(24,32)(25,28)(27,29);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,24,28)(19,30,31,20)(25,26,29,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6356853);