#10055822

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

5264 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)(9,10,14,16)(11,15,12,13)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32);(2,5)(6,7)(9,11)(10,15)(12,14)(13,16)(20,27)(24,30)(26,31)(28,32);(2,5)(6,7)(10,16)(11,12)(18,23)(19,25)(24,30)(26,31);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,13,16,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32);(1,2)(3,7)(4,5)(6,8)(9,12,14,11)(10,13,16,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32);(1,3,4,8)(2,6,5,7)(9,13)(10,12)(11,16)(14,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265951, #72265959, #72265967
maximal normal subgroup of	#72265951, #72265959, #72265967
maximal quotient of	#72266519, #72266562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9963018);