#7697480

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	#59824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4653 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=gh,e^2=g,f^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, 7604676);