#8456032

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

5181 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,e^2=gi,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, 8363228);