#8152213

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

5663 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,29)(22,26)(23,24)(25,30)(27,31)(28,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,24)(18,20,26,23)(21,27,30,32)(25,28,29,31);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(1,2,4,6)(3,5,8,7)(9,11,12,16)(10,13,14,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261815, #72261985, #72262208
maximal normal subgroup of	#72261815, #72261985, #72262208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059409);