#8093617

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

(1,2,6,7)(3,4,11,8)(5,9,13,14)(10,12,16,15)(18,20)(23,25)(24,28)(30,31)(33,34)(35,41)(36,47)(37,43)(38,53)(39,40)(42,51)(44,54)(45,49)(46,52)(48,55)(50,57)(56,58)(59,64)(60,63)(61,62);(2,7)(4,8)(9,14)(12,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(34,40)(37,41)(42,47)(44,53)(49,55)(52,56)(57,60)(62,64);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,39,43)(34,41,40,37)(36,48,51,45)(38,46,54,58)(42,49,47,55)(44,56,53,52)(50,61,63,59)(57,62,60,64);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,39,51)(34,42,40,47)(35,45,43,48)(37,49,41,55)(38,50,54,63)(44,57,53,60)(46,59,58,61)(52,62,56,64);(33,37,39,41)(34,43,40,35)(36,49,51,55)(38,52,54,56)(42,48,47,45)(44,58,53,46)(50,62,63,64)(57,61,60,59);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,38,39,54)(34,44,40,53)(35,46,43,58)(36,50,51,63)(37,52,41,56)(42,57,47,60)(45,59,48,61)(49,62,55,64)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000813);