#9054438

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	#42153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8961634);