#7697608

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	#35683	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4886 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,31)(28,32);(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,26,32)(20,27,30,21)(24,25,29,31);(2,5)(6,8)(17,20,22,30)(18,24,26,29)(19,21,28,27)(23,31,32,25);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,25)(26,29)(31,32);(1,3,4,7)(2,6,5,8)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,29,32,24);(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=h,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, 7604804);