#8729117

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

4904 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8636313);