#8757428

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

3636 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)(3,11)(4,14)(10,16)(19,25)(22,26)(27,31)(29,32)(33,34,39,44)(35,40,42,37)(36,41,50,56)(38,43,52,58)(45,53,47,48)(46,54,57,51)(49,55,62,64)(59,63,60,61);(1,2,7,6)(3,12,11,8)(4,9,14,15)(5,16,13,10)(17,18,23,20)(19,22,25,26)(21,28,30,24)(27,32,31,29)(35,42)(37,40)(38,52)(43,58)(45,47)(48,53)(49,62)(55,64);(1,3,7,11)(2,8,6,12)(4,13,14,5)(9,16,15,10)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,42)(34,40,44,37)(36,47,50,45)(38,46,52,57)(41,48,56,53)(43,54,58,51)(49,60,62,59)(55,61,64,63);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36)(34,41)(35,45)(37,48)(38,49)(39,50)(40,53)(42,47)(43,55)(44,56)(46,59)(51,61)(52,62)(54,63)(57,60)(58,64);(1,5,7,13)(2,10,6,16)(3,4,11,14)(8,9,12,15)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37,39,40)(34,42,44,35)(36,48,50,53)(38,51,52,54)(41,47,56,45)(43,57,58,46)(49,61,62,63)(55,60,64,59);(1,6,7,2)(3,12,11,8)(4,15,14,9)(5,16,13,10)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38)(34,43)(35,46)(36,49)(37,51)(39,52)(40,54)(41,55)(42,57)(44,58)(45,59)(47,60)(48,61)(50,62)(53,63)(56,64)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664624);