#8096149

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

5809 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304025, #72336231
maximal normal subgroup of	#72304025, #72336231
maximal quotient of	#72304047, #72304138, #72336412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003345);