#7632160

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

5058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539356);