#8197267

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

5889 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261815, #72262018, #72262049, …
maximal normal subgroup of	#72261815, #72262018, #72262049, …
maximal quotient of	#72300492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104463);