#7844610

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32);(1,2)(3,5)(4,6)(7,8)(17,19,22,29)(18,23,26,27)(20,25,30,31)(21,24,32,28);(10,13)(14,15)(17,18,22,26)(19,27,29,23)(20,24,30,28)(21,31,32,25);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32);(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,25)(21,27)(22,30)(23,32)(26,28)(29,31);(1,3,4,7)(2,5,6,8)(17,21,22,32)(18,25,26,31)(19,28,29,24)(20,27,30,23)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7751806);