#8926604

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	#54046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,39)(38,48)(40,49)(41,46)(42,50)(43,44)(45,47)(51,52)(53,55)(54,56);(1,2,6,8)(3,5,7,4)(13,21)(16,20)(18,24)(22,23)(26,32)(29,33)(34,46)(36,47)(39,42)(43,48)(49,56)(52,55);(1,3,6,7)(2,4,8,5)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,35)(26,33,32,29)(28,37,41,50)(30,44,45,38)(34,39,46,42)(36,43,47,48)(40,53,54,51)(49,55,56,52);(1,4,6,5)(2,3,8,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,34)(27,37)(29,42)(30,40)(31,41)(32,46)(33,39)(35,50)(36,49)(38,51)(43,55)(44,53)(45,54)(47,56)(48,52);(1,2)(3,4)(5,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,31,33)(26,35,32,27)(28,39,41,42)(30,43,45,48)(34,37,46,50)(36,44,47,38)(40,52,54,55)(49,51,56,53);(1,5,6,4)(2,7,8,3)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,30,31,45)(26,36,32,47)(27,38,35,44)(28,40,41,54)(29,43,33,48)(34,49,46,56)(37,51,50,53)(39,52,42,55)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=h,e^2=gh,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, 8833800);