#8159367

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5680 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)(9,10)(11,15)(12,14)(13,16)(21,29)(24,30)(26,31)(27,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(18,23)(19,25)(24,30)(26,31);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,12,15)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319676, #72319686, #72319764
maximal normal subgroup of	#72319676, #72319686, #72319764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066563);