#7709520

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	#39359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4828 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)(19,26)(20,21)(22,24)(23,27)(25,30)(28,32)(29,31);(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,31)(30,32);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(18,24)(19,27)(20,25)(29,32);(1,3,5,8)(2,6,4,7)(17,20,22,25)(18,21,24,30)(19,28,27,31)(23,29,26,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,31)(30,32);(17,21)(18,25)(19,29)(20,24)(22,30)(23,31)(26,28)(27,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,e^2=gh,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, 7616716);