#8934399

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	#60034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5143 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841595);