#8479034

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

5806 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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.

Relationships

maximal subgroup of	#72261815, #72261907
maximal normal subgroup of	#72261815, #72261907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386230);