#8209854

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

5785 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262049, #72262162
maximal normal subgroup of	#72262049, #72262162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8117050);