#8492801

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

5036 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)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(19,28)(21,29)(24,32)(25,30);(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32);(1,2)(3,5)(4,6)(7,8)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8399997);