#10261481

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

6105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=ghi,f^2=gi,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	#72326547, #72333952
maximal normal subgroup of	#72326547, #72333952
maximal quotient of	#72326639, #72333489

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10168677);