#10217787

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

5275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265951, #72265960, #72265967
maximal normal subgroup of	#72265951, #72265960, #72265967
maximal quotient of	#72266540, #72266566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10124983);