#8531708

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

6053 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261997, #72300326, #72340331
maximal normal subgroup of	#72261997, #72300326, #72340331
maximal quotient of	#72300398

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8438904);