#7696216

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	#35680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6058 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603412);