#7720364

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	#35236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,f^2=i,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, 7627560);