#8758096

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

5372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336552
maximal normal subgroup of	#72336552
maximal quotient of	#72336695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8665292);