#8434162

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

5883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,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, 8341358);