#6475274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	#37356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,21)(19,26,25,23)(20,28,29,27)(24,32,31,30);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,26)(22,25)(24,32)(27,29)(30,31);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,26,32,22,29,23,30)(18,24,19,27,21,31,25,28);(11,15)(13,16)(17,21,22,18)(19,26,25,23)(20,24,29,31)(27,30,28,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,21)(19,25)(20,29)(30,32)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei

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, 6382470);