#8607423

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

5143 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)(3,8)(4,5)(6,7)(9,10)(11,15)(12,14)(13,16);(1,3,6,5)(2,4,7,8)(9,10,12,14)(11,13,16,15)(17,18,22,25)(19,26,29,23)(20,21,24,30)(27,32,31,28);(2,7)(4,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,29)(18,23,25,26)(20,27,24,31)(21,28,30,32);(2,7)(3,5)(9,11)(10,13)(12,16)(14,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31);(1,4)(2,3)(5,7)(6,8)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32);(1,5,6,3)(2,4,7,8)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317637, #72319825, #72319886
maximal normal subgroup of	#72317637, #72319825, #72319886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8514619);