#7709455

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	#39503	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4860 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,28)(27,32)(29,31);(1,3,6,8)(2,7,4,5)(9,10,13,14)(11,15,12,16)(18,23)(19,25)(24,31)(26,27);(2,4)(3,8)(9,11,13,12)(10,15,14,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28);(1,3,6,8)(2,7,4,5);(1,4,6,2)(3,5,8,7)(9,12,13,11)(10,16,14,15)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32);(1,5)(2,3)(4,8)(6,7)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,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, 7616651);