#8429644

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

6296 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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.

Relationships

maximal subgroup of	#72261791, #72261968
maximal normal subgroup of	#72261791, #72261968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336840);