#8093613

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	#53151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000809);