#8726831

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

6054 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262018, #72262216
maximal normal subgroup of	#72262018, #72262216
maximal quotient of	#72300475

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8634027);