#7642119

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

4011 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549315);