#8757342

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

3707 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:

(2,4)(3,10)(6,14)(13,15)(18,22)(20,27)(23,28)(26,31)(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);(1,2,7,4)(3,5,10,8)(6,9,14,12)(11,13,16,15)(19,25)(23,28)(26,31)(30,32)(35,42)(37,40)(38,52)(43,58)(45,47)(48,53)(49,62)(55,64);(1,3,7,10)(2,8,4,5)(6,11,14,16)(9,15,12,13)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(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);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(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)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(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)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(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=g,c^2=i,d^2=g,e^2=gi,f^2=gh,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, 8664538);