#7714789

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^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, 7621985);