#8891375

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798571);