#8677327

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#36125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8584523);