#8926577

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	#54045	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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,34)(27,31,37,46)(28,32,41,39)(29,33,43,50)(35,44,38,47)(36,45,42,49)(40,48,54,53)(51,55,52,56);(2,5)(3,8)(12,20)(15,22)(18,24)(21,23)(25,27,30,37)(26,31,34,46)(28,38,41,35)(29,36,43,42)(32,47,39,44)(33,45,50,49)(40,52,54,51)(48,56,53,55);(1,3)(2,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,35)(29,42)(30,37)(32,44)(33,49)(34,46)(36,43)(38,41)(39,47)(40,52)(45,50)(48,56)(51,54)(53,55);(1,4,7,6)(2,3,5,8)(25,26)(27,31)(28,39)(29,33)(30,34)(32,41)(35,47)(36,45)(37,46)(38,44)(40,53)(42,49)(43,50)(48,54)(51,56)(52,55);(1,5,7,2)(3,4,8,6)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,32)(27,35)(29,40)(30,41)(31,44)(33,48)(34,39)(36,51)(37,38)(42,52)(43,54)(45,55)(46,47)(49,56)(50,53);(1,6,7,4)(2,8,5,3)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29)(26,33)(27,36)(28,40)(30,43)(31,45)(32,48)(34,50)(35,51)(37,42)(38,52)(39,53)(41,54)(44,55)(46,49)(47,56)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

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, 8833773);