#7737475

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5217 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644671);