#7658828

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566024);