#6490818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	#35303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,22,23,21,25,20,26)(19,24,29,30,27,32,28,31);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(18,23)(19,27)(20,22)(24,31)(25,26)(30,32);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,19,21,27)(18,24,25,32)(20,28,22,29)(23,30,26,31);(9,12,14,13)(10,15,11,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,25)(19,27)(23,26)(28,29)

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398014);