#6459199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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)(11,14)(12,13)(15,16)(17,18,22,23,21,26,20,25)(19,24,29,30,28,31,27,32);(2,6)(7,8)(18,23)(19,27)(20,22)(25,26)(28,29)(30,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,22,29,21,28,20,27)(18,24,23,30,26,31,25,32);(1,4,5,3)(2,8,6,7)(17,19,20,27,21,28,22,29)(18,24,25,32,26,31,23,30);(9,11,12,15)(10,14,13,16)(19,28)(24,31)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,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, 6366395);