#7715683

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622879);