#8127504

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

5896 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)(3,5)(4,7)(6,8)(10,12)(14,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,32)(27,31)(28,29);(2,3)(6,7)(11,15)(14,16)(17,19,23,21)(18,24,25,30)(20,29,26,27)(22,28,32,31);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(18,25)(19,21)(20,26)(28,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,20,23,26)(18,22,25,32)(19,27,21,29)(24,31,30,28);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,21,23,19)(18,24,25,30)(20,29,26,27)(22,31,32,28);(17,22)(18,26)(19,28)(20,25)(21,31)(23,32)(24,27)(29,30)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=h,e^2=h,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^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312548, #72342337
maximal normal subgroup of	#72312548, #72342337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034700);