#7644274

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

4043 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)(9,10,14,18)(11,19,20,15)(12,16,13,17)(21,24,22,23)(29,41)(34,43)(37,48)(39,52)(45,53)(46,51)(50,55)(54,56);(1,2,4,5)(3,6,7,8)(11,20)(15,19)(21,22)(23,24)(26,31)(27,35)(33,38)(34,43)(36,47)(37,48)(46,51)(50,55);(9,11,14,20)(10,15,18,19)(12,21,13,22)(16,23,17,24)(25,26,30,31)(27,32,35,42)(28,38,40,33)(29,34,41,43)(36,49,47,44)(37,45,48,53)(39,51,52,46)(50,56,55,54);(1,3,4,7)(2,6,5,8)(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)(33,44)(34,45)(38,49)(39,50)(40,47)(41,48)(43,53)(46,54)(51,56)(52,55);(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,23,19,24)(25,28,30,40)(26,33,31,38)(27,36,35,47)(29,39,41,52)(32,44,42,49)(34,46,43,51)(37,50,48,55)(45,54,53,56);(9,13,14,12)(10,17,18,16)(11,22,20,21)(15,24,19,23)(25,29)(26,34)(27,37)(28,39)(30,41)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(40,52)(42,53)(44,54)(47,55)(49,56)

Polycyclic presentation:

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

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, 7551470);