#8689970

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

5765 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312311, #72342330, #72342337
maximal normal subgroup of	#72312311, #72342330, #72342337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8597166);