#7696222

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	#45103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5148 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=gh,f^2=gh,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, 7603418);