#8115425

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

3735 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:

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

Polycyclic presentation:

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

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, 8022621);