#8456847

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

5095 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8364043);