#7719375

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

4654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7626571);