#6495969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	#25508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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)(4,6)(5,7)(11,15)(14,16)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,19,21,29,22,28,32,24)(18,23,26,27,25,20,30,31);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(19,28)(20,23)(24,29)(27,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,24)(19,26)(21,31)(22,23)(25,29)(27,32)(28,30);(9,11,12,15)(10,14,13,16)(18,25)(20,23)(26,30)(27,31)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6403165);