#7709461

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	#37679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4846 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=h,f^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, 7616657);