#7715465

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

3956 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)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,26,31,28)(27,35,36,32)(29,39,43,33)(30,40,44,34)(37,45,41,47)(38,46,50,49)(42,48,54,52)(51,56,53,55);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(26,28)(27,36)(33,39)(34,40)(37,41)(38,50)(48,52)(51,53);(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,27,31,36)(26,32,28,35)(29,41,43,37)(30,38,44,50)(33,47,39,45)(34,46,40,49)(42,53,54,51)(48,56,52,55);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,28,31,26)(27,32,36,35)(29,39,43,33)(30,40,44,34)(37,45,41,47)(38,46,50,49)(42,52,54,48)(51,55,53,56);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29)(26,33)(27,37)(28,39)(30,42)(31,43)(32,45)(34,48)(35,47)(36,41)(38,51)(40,52)(44,54)(46,55)(49,56)(50,53);(25,30,31,44)(26,34,28,40)(27,38,36,50)(29,42,43,54)(32,46,35,49)(33,48,39,52)(37,51,41,53)(45,55,47,56)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,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, 7622661);