#8496823

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

4974 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,f^2=gi,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, 8404019);