#8688942

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	#43893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596138);