#8757279

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	#45612	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=i,f^2=g,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, 8664475);