#7709281

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	#36126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4812 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616477);