#8677357

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	#36471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4587 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 8584553);