#7737499

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

5218 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:

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

Polycyclic presentation:

2^9|a^2=g,c^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, 7644695);