#7694174

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	#52160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4576 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18)(19,24)(20,22)(21,27)(23,25)(26,29)(28,30)(31,32);(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,21)(22,27)(23,28)(25,30)(26,32)(29,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,25)(19,28)(20,29)(27,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,20)(18,26)(19,21)(22,25)(23,29)(24,32)(27,30)(28,31);(1,3,4,7)(2,6,5,8)(17,21)(18,27)(19,20)(22,24)(23,31)(25,32)(26,30)(28,29);(17,22,23,26)(18,20,25,29)(19,27,28,32)(21,30,31,24)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=hi,f^2=i,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, 7601370);