#8932529

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	#51324	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839725);