#8161188

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

5068 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)(10,13)(14,16)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28);(2,5)(6,7)(11,15)(14,16)(18,24)(20,25)(26,30)(28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29);(1,3,4,8)(2,6,5,7)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32);(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311840
maximal normal subgroup of	#72311840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8068384);