#10480708

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

5303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265954, #72265960, #72265962
maximal normal subgroup of	#72265954, #72265960, #72265962
maximal quotient of	#72266510, #72266558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10387904);