#8159339

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

5734 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^2=g,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	#72319650, #72319676, #72319764
maximal normal subgroup of	#72319650, #72319676, #72319764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066535);