#8347965

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

5976 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)(3,6)(4,5)(7,8)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,32)(30,31);(2,5)(6,8)(9,10)(11,15)(12,13)(14,16)(18,23)(19,26)(20,29)(31,32);(3,7)(6,8)(10,13)(14,15)(18,23)(21,24)(25,28)(31,32);(2,5)(6,8)(11,16)(14,15)(17,19)(18,24)(20,30)(21,23)(22,26)(25,31)(27,29)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,27)(21,31)(22,29)(23,28)(24,32)(26,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,24)(18,19,23,26)(20,31,29,32)(25,27,28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261815, #72262018
maximal normal subgroup of	#72261815, #72262018
maximal quotient of	#72300516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8255161);