#7782149

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	#34963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4961 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,f^2=gi,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, 7689345);