#8924388

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

5448 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)(3,5,7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,25)(20,30)(21,23)(22,26)(24,27)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(10,14)(12,15)(17,19,24,21)(18,25,27,23)(20,29,22,28)(26,32,30,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20,24,22)(18,26,27,30)(19,28,21,29)(23,32,25,31);(9,11)(10,12)(13,16)(14,15)(17,21,24,19)(18,23,27,25)(20,28,22,29)(26,31,30,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,22,24,20)(18,26,27,30)(19,29,21,28)(23,32,25,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,23)(18,19)(20,31)(21,27)(22,32)(24,25)(26,29)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219287
maximal normal subgroup of	#72219285, #72219287
maximal quotient of	#72219276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831584);