#7683183

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

4784 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)(3,6)(4,8)(5,7)(10,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,15,12,16)(17,19,23,27)(18,24,22,30)(20,28,29,21)(25,32,31,26);(2,7)(6,8)(9,11,13,12)(10,15,14,16)(18,22)(19,27)(20,29)(26,32);(1,2)(3,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,27)(22,31)(23,29)(24,32)(26,30);(17,21,23,28)(18,26,22,32)(19,29,27,20)(24,31,30,25);(1,4,5,3)(2,8,7,6)(9,12,13,11)(10,16,14,15)(17,22,23,18)(19,30,27,24)(20,31,29,25)(21,32,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590379);