#10320346

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10227542);