#8108013

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4572 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:

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

Polycyclic presentation:

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

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, 8015209);