#8612175

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

4856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=gi,f^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8519371);