#7768086

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

4928 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7675282);